This problem arises when searching the particular solution of the differential equation. Our online calculator, based on the Wolfram Alpha system allows you to find a solution of Cauchy problem for various types of differential equations. To get started, you need to enter your task's data (differential equation, initial conditions) in the calculator.
When setting the Cauchy problem, the so-called initial conditions are specified, which allow us to uniquely distinguish the desired particular solution from the general one. These conditions include the values of the functions and all its derivatives up to inclusively (where - is the order of the differential equation), given at the same point .
Consider the following example. Assume we need to find a particular solution to the differential equation:
satisfying the initial conditions:
First of all, by using various methods (Bernoulli, variation of an arbitrary Lagrange constant), we find a general solution to this differential equation:
Now, to find a particular solution, we need to use the specified initial conditions. To do this, find the derivative of the function obtained earlier:
Next, substitute the initial conditions into the function and its derivative :
Solving the resulting system of equations, we obtain the values of arbitrary constants and :
Substituting the obtained results into a general solution of the differential equation, we find the desired particular solution: