This online calculator allows you to solve differential equations online. It is enough to enter your equation in the appropriate field, denoting the derivative of the function through an apostrophe, and click on the “solve equation” button. And the system, implemented on the basis of the popular WolframAlpha website, will give detailed solving a differential equation absolutely free. You can also define a Cauchy problem to select from the entire set of possible solutions the quotient that corresponds to the given initial conditions. The Cauchy problem is entered in a separate field.
Differential equation
By default, the function in the equation y is a function of a variable x. However, you can specify your own designation for the variable; if you write, for example, y(t) in the equation, the calculator will automatically recognize that y there is a function from a variable t. With the help of a calculator you can solve differential equations of any complexity and type: homogeneous and inhomogeneous, linear or nonlinear, first order or second and higher orders, equations with separable or nonseparable variables, etc. Solution diff. the equation is given in analytical form and has a detailed description. Differential equations are very common in physics and mathematics. Without calculating them, it is impossible to solve many problems (especially in mathematical physics).
One of the stages of solving differential equations is integrating functions. There are standard methods for solving differential equations. It is necessary to reduce the equations to a form with separable variables y and x and separately integrate the separated functions. To do this, sometimes a certain replacement must be made.
Solving differential equations. Thanks to our online service, you can solve differential equations of any type and complexity: inhomogeneous, homogeneous, nonlinear, linear, first, second order, with separable or non-separable variables, etc. You receive a solution to differential equations in analytical form with a detailed description. Many people are interested: why is it necessary to solve differential equations online? This type of equation is very common in mathematics and physics, where it will be impossible to solve many problems without calculating the differential equation. Differential equations are also common in economics, medicine, biology, chemistry and other sciences. Solving such an equation online greatly simplifies your tasks, gives you the opportunity to better understand the material and test yourself. Advantages of solving differential equations online. A modern mathematical service website allows you to solve differential equations online of any complexity. As you know, there are a large number of types of differential equations and each of them has its own methods of solution. On our service you can find solutions to differential equations of any order and type online. To get a solution, we suggest you fill in the initial data and click the “Solution” button. Errors in the operation of the service are excluded, so you can be 100% sure that you received the correct answer. Solve differential equations with our service. Solve differential equations online. By default, in such an equation, the function y is a function of the x variable. But you can also specify your own variable designation. For example, if you specify y(t) in a differential equation, then our service will automatically determine that y is a function of the t variable. The order of the entire differential equation will depend on the maximum order of the derivative of the function present in the equation. Solving such an equation means finding the desired function. Our service will help you solve differential equations online. It doesn't take much effort on your part to solve the equation. You just need to enter the left and right sides of your equation into the required fields and click the “Solution” button. When entering, the derivative of a function must be denoted by an apostrophe. In a matter of seconds you will receive a ready-made detailed solution to the differential equation. Our service is absolutely free. Differential equations with separable variables. If in a differential equation there is an expression on the left side that depends on y, and on the right side there is an expression that depends on x, then such a differential equation is called with separable variables. The left side may contain a derivative of y; the solution to differential equations of this type will be in the form of a function of y, expressed through the integral of the right side of the equation. If on the left side there is a differential of the function of y, then in this case both sides of the equation are integrated. When the variables in a differential equation are not separated, they will need to be separated to obtain a separated differential equation. Linear differential equation. A differential equation whose function and all its derivatives are in the first degree is called linear. General form of the equation: y’+a1(x)y=f(x). f(x) and a1(x) are continuous functions of x. Solving differential equations of this type reduces to integrating two differential equations with separated variables. Order of differential equation. A differential equation can be of the first, second, nth order. The order of a differential equation determines the order of the highest derivative that it contains. In our service you can solve differential equations online for the first, second, third, etc. order. The solution to the equation will be any function y=f(x), substituting it into the equation, you will get an identity. The process of finding a solution to a differential equation is called integration. Cauchy problem. If, in addition to the differential equation itself, the initial condition y(x0)=y0 is given, then this is called the Cauchy problem. The indicators y0 and x0 are added to the solution of the equation and the value of an arbitrary constant C is determined, and then a particular solution of the equation at this value of C is determined. This is the solution to the Cauchy problem. The Cauchy problem is also called a problem with boundary conditions, which is very common in physics and mechanics. You also have the opportunity to set the Cauchy problem, that is, from all possible solutions to the equation, select a quotient that meets the given initial conditions.
Let us recall the task that confronted us when finding definite integrals:
or dy = f(x)dx. Her solution:
and it comes down to calculating the indefinite integral. In practice, a more complex task is more often encountered: finding the function y, if it is known that it satisfies a relation of the form
This relationship relates the independent variable x, unknown function y and its derivatives up to the order n inclusive, are called .
A differential equation includes a function under the sign of derivatives (or differentials) of one order or another. The highest order is called order (9.1) .
Differential equations:
- first order,
Second order
- fifth order, etc.
The function that satisfies a given differential equation is called its solution , or integral . Solving it means finding all its solutions. If for the required function y managed to obtain a formula that gives all solutions, then we say that we have found its general solution , or general integral .
Common decision
contains n arbitrary constants 
and looks like
If a relation is obtained that relates x, y And n arbitrary constants, in a form not permitted with respect to y -
then such a relation is called the general integral of equation (9.1).
Cauchy problem
Each specific solution, i.e., each specific function that satisfies a given differential equation and does not depend on arbitrary constants, is called a particular solution , or a partial integral. To obtain particular solutions (integrals) from general ones, the constants must be given specific numerical values.
The graph of a particular solution is called an integral curve. The general solution, which contains all the partial solutions, is a family of integral curves. For a first-order equation this family depends on one arbitrary constant, for the equation n-th order - from n arbitrary constants.
The Cauchy problem is to find a particular solution for the equation n-th order, satisfying n initial conditions:
by which n constants c 1, c 2,..., c n are determined.
1st order differential equations
For a 1st order differential equation that is unresolved with respect to the derivative, it has the form
or for permitted relatively
Example 3.46. Find the general solution to the equation
Solution. Integrating, we get
where C is an arbitrary constant. If we assign specific numerical values to C, we obtain particular solutions, for example,
Example 3.47. Consider an increasing amount of money deposited in the bank subject to the accrual of 100 r compound interest per year. Let Yo be the initial amount of money, and Yx - at the end x years. If interest is calculated once a year, we get
where x = 0, 1, 2, 3,.... When interest is calculated twice a year, we get
where x = 0, 1/2, 1, 3/2,.... When calculating interest n once a year and if x takes sequential values 0, 1/n, 2/n, 3/n,..., then
Designate 1/n = h, then the previous equality will look like:
With unlimited magnification n(at 
) in the limit we come to the process of increasing the amount of money with continuous accrual of interest:
Thus it is clear that with continuous change x the law of change in the money supply is expressed by a 1st order differential equation. Where Y x is an unknown function, x- independent variable, r- constant. Let's solve this equation, to do this we rewrite it as follows:
where 
, or 
, where P denotes e C .
From the initial conditions Y(0) = Yo, we find P: Yo = Pe o, from where, Yo = P. Therefore, the solution has the form:
Let's consider the second economic problem. Macroeconomic models are also described by linear differential equations of the 1st order, describing changes in income or output Y as functions of time.
Example 3.48. Let national income Y increase at a rate proportional to its value:
and let the deficit in government spending be directly proportional to income Y with the proportionality coefficient q. A spending deficit leads to an increase in national debt D:
Initial conditions Y = Yo and D = Do at t = 0. From the first equation Y= Yoe kt. Substituting Y we get dD/dt = qYoe kt . The general solution has the form
D = (q/ k) Yoe kt +С, where С = const, which is determined from the initial conditions. Substituting the initial conditions, we get Do = (q/ k)Yo + C. So, finally,
D = Do +(q/ k)Yo (e kt -1),
this shows that the national debt is increasing at the same relative rate k, the same as national income.
Let us consider the simplest differential equations n th order, these are equations of the form
Its general solution can be obtained using n times integrations.
Example 3.49. Consider the example y """ = cos x.
Solution. Integrating, we find
The general solution has the form
Linear differential equations
They are widely used in economics; let’s consider solving such equations. If (9.1) has the form:
then it is called linear, where рo(x), р1(x),..., рn(x), f(x) are given functions. If f(x) = 0, then (9.2) is called homogeneous, otherwise it is called inhomogeneous. The general solution of equation (9.2) is equal to the sum of any of its particular solutions y(x) and the general solution of the homogeneous equation corresponding to it:
If the coefficients р o (x), р 1 (x),..., р n (x) are constant, then (9.2)
(9.4) is called a linear differential equation with constant coefficients of order n .
For (9.4) has the form:
Without loss of generality, we can set p o = 1 and write (9.5) in the form
We will look for a solution (9.6) in the form y = e kx, where k is a constant. We have: ; y " = ke kx , y "" = k 2 e kx , ..., y (n) = kne kx . Substituting the resulting expressions into (9.6), we will have:
(9.7) is an algebraic equation, its unknown is k, it is called characteristic. The characteristic equation has degree n And n roots, among which there can be both multiple and complex. Let k 1 , k 2 ,..., k n be real and distinct, then 
- particular solutions (9.7), and general
Consider a linear homogeneous second-order differential equation with constant coefficients:
Its characteristic equation has the form
(9.9)
its discriminant D = p 2 - 4q, depending on the sign of D, three cases are possible.
1. If D>0, then the roots k 1 and k 2 (9.9) are real and different, and the general solution has the form:
Solution. Characteristic equation: k 2 + 9 = 0, whence k = ± 3i, a = 0, b = 3, the general solution has the form:
y = C 1 cos 3x + C 2 sin 3x.
Linear differential equations of the 2nd order are used when studying a web-type economic model with inventories of goods, where the rate of change in price P depends on the size of the inventory (see paragraph 10). If supply and demand are linear functions of price, that is
a is a constant that determines the reaction rate, then the process of price change is described by the differential equation:
For a particular solution we can take a constant
meaningful equilibrium price. Deviation 
satisfies the homogeneous equation
(9.10)
The characteristic equation will be as follows:
In case the term is positive. Let's denote 
. The roots of the characteristic equation k 1,2 = ± i w, therefore the general solution (9.10) has the form:
where C and are arbitrary constants, they are determined from the initial conditions. We obtained the law of price change over time:
6.1. BASIC CONCEPTS AND DEFINITIONS
When solving various problems in mathematics and physics, biology and medicine, quite often it is not possible to immediately establish a functional relationship in the form of a formula connecting the variables that describe the process under study. Usually you have to use equations that contain, in addition to the independent variable and the unknown function, also its derivatives.
Definition. An equation connecting an independent variable, an unknown function and its derivatives of various orders is called differential.
An unknown function is usually denoted y(x) or simply y, and its derivatives - y", y" etc.
Other designations are also possible, for example: if y= x(t), then x"(t), x""(t)- its derivatives, and t- independent variable.
Definition. If a function depends on one variable, then the differential equation is called ordinary. General form ordinary differential equation:
or
Functions F And f may not contain some arguments, but for the equations to be differential, the presence of a derivative is essential.
Definition.The order of the differential equation is called the order of the highest derivative included in it.
For example, x 2 y"- y= 0, y" + sin x= 0 are first order equations, and y"+ 2 y"+ 5 y= x- second order equation.
When solving differential equations, the integration operation is used, which is associated with the appearance of an arbitrary constant. If the integration action is applied n times, then, obviously, the solution will contain n arbitrary constants.
6.2. DIFFERENTIAL EQUATIONS OF THE FIRST ORDER
General form first order differential equation is determined by the expression
The equation may not explicitly contain x And y, but necessarily contains y".
If the equation can be written as
then we obtain a first-order differential equation resolved with respect to the derivative.
Definition. The general solution of the first order differential equation (6.3) (or (6.4)) is the set of solutions 
, Where WITH- arbitrary constant.
The graph of the solution to a differential equation is called integral curve.
Giving an arbitrary constant WITH different values, partial solutions can be obtained. On surface xOy the general solution is a family of integral curves corresponding to each particular solution.
If you set a point A (x 0 , y 0), through which the integral curve must pass, then, as a rule, from a set of functions 
One can single out one - a private solution.
Definition.Private decision of a differential equation is its solution that does not contain arbitrary constants.
If 
is a general solution, then from the condition
you can find a constant WITH. The condition is called initial condition.
The problem of finding a particular solution to the differential equation (6.3) or (6.4) satisfying the initial condition 
at 
called Cauchy problem. Does this problem always have a solution? The answer is contained in the following theorem.
Cauchy's theorem(theorem of existence and uniqueness of a solution). Let in the differential equation y"= f(x,y) function f(x,y) and her
partial derivative 
defined and continuous in some
region D, containing a point 
Then in the area D exists
the only solution to the equation that satisfies the initial condition 
at 
Cauchy's theorem states that under certain conditions there is a unique integral curve y= f(x), passing through a point 
Points at which the conditions of the theorem are not met
Cauchies are called special. At these points it breaks f(x, y) or.
Either several integral curves or none pass through a singular point.
Definition. If the solution (6.3), (6.4) is found in the form f(x, y, C)= 0, not allowed relative to y, then it is called general integral differential equation.
Cauchy's theorem only guarantees that a solution exists. Since there is no single method for finding a solution, we will consider only some types of first-order differential equations that can be integrated into quadratures
Definition. The differential equation is called integrable in quadratures, if finding its solution comes down to integrating functions.
6.2.1. First order differential equations with separable variables
Definition. A first order differential equation is called an equation with separable variables,
The right side of equation (6.5) is the product of two functions, each of which depends on only one variable.
For example, the equation 
is an equation with separating
mixed with variables 
and the equation 
cannot be represented in the form (6.5).
Considering that 
, we rewrite (6.5) in the form 
From this equation we obtain a differential equation with separated variables, in which the differentials are functions that depend only on the corresponding variable:
Integrating term by term, we have
where C = C 2 - C 1 - arbitrary constant. Expression (6.6) is the general integral of equation (6.5).
By dividing both sides of equation (6.5) by, we can lose those solutions for which, 
Indeed, if 
at 
That 
obviously is a solution to equation (6.5).
Example 1. Find a solution to the equation that satisfies
condition: y= 6 at x= 2 (y(2) = 6).
Solution. We will replace y" then 
. Multiply both sides by
dx, since during further integration it is impossible to leave dx in the denominator:
and then dividing both parts by 
we get the equation,
which can be integrated. Let's integrate: 
Then 
; potentiating, we get y = C. (x + 1) - ob-
general solution.
Using the initial data, we determine an arbitrary constant, substituting them into the general solution
Finally we get y= 2(x + 1) is a particular solution. Let's look at a few more examples of solving equations with separable variables.
Example 2. Find the solution to the equation 
Solution. Considering that 
, we get 
.
Integrating both sides of the equation, we have
where 
Example 3. Find the solution to the equation Solution. We divide both sides of the equation into those factors that depend on a variable that does not coincide with the variable under the differential sign, i.e. 
and integrate. Then we get
and finally
Example 4. Find the solution to the equation
Solution. Knowing what we will get. Section
lim variables. Then 
Integrating, we get
Comment. In examples 1 and 2, the required function is y expressed explicitly (general solution). In examples 3 and 4 - implicitly (general integral). In the future, the form of the decision will not be specified.
Example 5. Find the solution to the equation Solution.
Example 6. Find the solution to the equation 
, satisfying
condition y(e)= 1.
Solution. Let's write the equation in the form 
Multiplying both sides of the equation by dx and on, we get
Integrating both sides of the equation (the integral on the right side is taken by parts), we obtain 
But according to the condition y= 1 at x= e. Then
Let's substitute the found values WITH to the general solution:
The resulting expression is called a partial solution of the differential equation.
6.2.2. Homogeneous differential equations of the first order
Definition. The first order differential equation is called homogeneous, if it can be represented in the form
Let us present an algorithm for solving a homogeneous equation.
1.Instead y let's introduce a new functionThen 
and therefore 
2.In terms of function u equation (6.7) takes the form
that is, the replacement reduces a homogeneous equation to an equation with separable variables.
3. Solving equation (6.8), we first find u and then y= ux.
Example 1. Solve the equation 
Solution. Let's write the equation in the form
We make the substitution: 
Then 
We will replace 
Multiply by dx: 
Divide by x and on 
Then
Having integrated both sides of the equation over the corresponding variables, we have
or, returning to the old variables, we finally get
Example 2.Solve the equation 
Solution.Let 
Then
Let's divide both sides of the equation by x2: 
Let's open the brackets and rearrange the terms:
Moving on to the old variables, we arrive at the final result:
Example 3.Find the solution to the equation 
given that
Solution.Performing a standard replacement 
we get
or 
or
This means that the particular solution has the form 
Example 4. Find the solution to the equation
Solution.
Example 5.Find the solution to the equation 
Solution.
Independent work
Find solutions to differential equations with separable variables (1-9).
Find a solution to homogeneous differential equations (9-18).
6.2.3. Some applications of first order differential equations
Radioactive decay problem
The rate of decay of Ra (radium) at each moment of time is proportional to its available mass. Find the law of radioactive decay of Ra if it is known that at the initial moment there was Ra and the half-life of Ra is 1590 years.
Solution. Let at the instant the mass Ra be x= x(t) g, and 
Then the decay rate Ra is equal to
According to the conditions of the problem 
Where k
Separating the variables in the last equation and integrating, we get
where 
For determining C we use the initial condition: when 
.
Then 
and, therefore, 
Proportionality factor k determined from the additional condition:
We have
From here 
and the required formula
Bacterial reproduction rate problem
The rate of reproduction of bacteria is proportional to their number. At the beginning there were 100 bacteria. Within 3 hours their number doubled. Find the dependence of the number of bacteria on time. How many times will the number of bacteria increase within 9 hours?
Solution. Let x- number of bacteria at a time t. Then, according to the condition, 
Where k- proportionality coefficient.
From here 
From the condition it is known that 
. Means,
From the additional condition 
. Then
The function you are looking for: 
So, when t= 9 x= 800, i.e. within 9 hours the number of bacteria increased 8 times.
The problem of increasing the amount of enzyme
In a brewer's yeast culture, the rate of growth of the active enzyme is proportional to its initial amount x. Initial amount of enzyme a doubled within an hour. Find dependency
x(t).
Solution. By condition, the differential equation of the process has the form 
from here
But 
. Means, C= a and then 
It is also known that 
Hence,
6.3. SECOND ORDER DIFFERENTIAL EQUATIONS
6.3.1. Basic Concepts
Definition.Second order differential equation is called a relation connecting the independent variable, the desired function and its first and second derivatives.
In special cases, x may be missing from the equation, at or y". However, a second-order equation must necessarily contain y." In the general case, a second-order differential equation is written as:
or, if possible, in the form resolved with respect to the second derivative:
As in the case of a first-order equation, for a second-order equation there can be general and particular solutions. The general solution is:
Finding a Particular Solution
under initial conditions - given
numbers) is called Cauchy problem. Geometrically, this means that we need to find the integral curve at= y(x), passing through a given point 
and having a tangent at this point which is
aligns with the positive axis direction Ox specified angle. e. 
(Fig. 6.1). The Cauchy problem has a unique solution if the right-hand side of equation (6.10), 
incessant
is discontinuous and has continuous partial derivatives with respect to uh, uh" in some neighborhood of the starting point
To find constants 
included in a private solution, the system must be resolved
Rice. 6.1. Integral curve
