Linear Differential Equations Calculator

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Example

Created on 2024-06-20Asked by Mason Adams (Solvelet student)

Solve the linear differential equation

\frac{dy}{dx} + y = e^x

Solution

To solve the linear differential equation

\frac{dy}{dx} + y = e^x

, we use the integrating factor method. The differential equation is in the standard form

\frac{dy}{dx} + P(x)y = Q(x)

, with

P(x) = 1

and

Q(x) = e^x

. First, find the integrating factor

\mu(x)

\mu(x) = e^{\int P(x) \, dx} = e^{\int 1 \, dx} = e^x

Multiply both sides of the differential equation by

\mu(x)

e^x \frac{dy}{dx} + e^x y = e^x e^x

This simplifies to:

e^x \frac{dy}{dx} + e^x y = e^{2x}

The left side is the derivative of

e^x y

\frac{d}{dx} (e^x y) = e^{2x}

Integrate both sides with respect to

x

e^x y = \int e^{2x} \, dx = \frac{1}{2} e^{2x} + C

Solve for

y

y = e^{-x} \left( \frac{1}{2} e^{2x} + C \right) = \frac{1}{2} e^x + C e^{-x}

Therefore, the general solution is:

y = \frac{1}{2} e^x + C e^{-x}

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Elijah Baker on Solvelet

1. Solve the linear differential equation

\frac{{dy}}{{dx}} + 3y = 2x

.2. Find the particular solution to the linear differential equation

y'' - y = x^2

DefinitionA differential equation that can be expressed in this form is called a linear differential equation, where y is the function to solve the equation for. The terms of y and its derivatives in the equation, which are in a linear polynomial, are called the coefficients. d I n t e g r a t i o n d x d y + P  y = Q a s i n S o l = ∫ d I n t e g r a t i o n + d I n t e g r a t i o n + d I n t e g r a t i o n + ∫ Q d x d I n t e g r a t i o n ∀ x ∈ R \mathbb{R} E x a m p l e A l i n e x a e q u a t i o n i s a d x d y + 3 y = 6

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