Particular Solutions Calculator

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Example

Created on 2024-06-20Asked by Amelia Jackson (Solvelet student)

Find the particular solution to the differential equation

\frac{dy}{dx} = 2x

with the initial condition

y(0) = 3

Solution

To find the particular solution to the differential equation

\frac{dy}{dx} = 2x

with the initial condition

y(0) = 3

: 1. Integrate both sides of the equation:

\int \frac{dy}{dx} \, dx = \int 2x \, dx.

2. Perform the integration:

y = x^2 + C,

where

C

is the constant of integration. 3. Apply the initial condition

y(0) = 3

3 = 0^2 + C,

C = 3.

4. Substitute the value of

C

back into the equation:

y = x^2 + 3.

Therefore, the particular solution to the differential equation is

y = x^2 + 3

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Levi Lopez on Solvelet

1. Find the particular solution to the differential equation

y'' + 2y' + y = 4

with the initial conditions

y(0) = 1

and

y'(0) = 2

.2. Verify whether the function

y = e^{2x}

is a particular solution to the differential equation

y'' - 4y' + 4y = 0

DefinitionA unique solution which satisfies the differential equation and some specified initial or boundary conditions is called a particular solution of a differential equation. The Initial or boundary conditions are enforced on the general solution to get the particular solution. Example: Since the differential equation y′′+y=0 with the initial conditions y(0)=1 and y′(0)=0 has the particular solution y=cos(x).

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