# Linear Differential Equation Calculator

## Get detailed solutions to your math problems with our Linear Differential Equation step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here.

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###  Difficult Problems

1

Solved example of linear differential equation

$y'+2y-3x=0$
2

Rewrite the differential equation using Leibniz notation

$\frac{dy}{dx}+2y-3x=0$
3

We can identify that the differential equation has the form: $\frac{dy}{dx} + P(x)\cdot y(x) = Q(x)$, so we can classify it as a linear first order differential equation, where $P(x)=2$ and $Q(x)=0$. In order to solve the differential equation, the first step is to find the integrating factor $\mu(x)$

$\displaystyle\mu\left(x\right)=e^{\int P(x)dx}$

Compute the integral

$\int2dx$

The integral of a constant is equal to the constant times the integral's variable

$2x$
4

To find $\mu(x)$, we first need to calculate $\int P(x)dx$

$\int P(x)dx=\int2dx=2x$
5

So the integrating factor $\mu(x)$ is

$\mu(x)=e^{2x}$

Any expression multiplied by $0$ is equal to $0$

$e^{2x}\frac{dy}{dx}+2ye^{2x}-3x=0$
6

Now, multiply all the terms in the differential equation by the integrating factor $\mu(x)$ and check if we can simplify

$e^{2x}\frac{dy}{dx}+2ye^{2x}-3x=0$
7

We can recognize that the left side of the differential equation consists of the derivative of the product of $\mu(x)\cdot y(x)$

$\frac{d}{dx}\left(e^{2x}y\right)=0$
8

Integrate both sides of the differential equation with respect to $dx$

$\int\frac{d}{dx}\left(e^{2x}y\right)dx=\int0dx$
9

Simplify the left side of the differential equation

$e^{2x}y=\int0dx$

The integral of a constant is equal to the constant times the integral's variable

0
10

Solve the integral $\int0dx$ and replace the result in the differential equation

$e^{2x}y=0$
11

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$e^{2x}y=0+C_0$
12

$x+0=x$, where $x$ is any expression

$e^{2x}y=C_0$

Multiply the equation by the reciprocal of $e^{2x}$

$y=e^{-2x}C_0$
13

Find the explicit solution to the differential equation

$y=e^{-2x}C_0$

$y=e^{-2x}C_0$