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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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Example

Solved Problems

Difficult Problems

1

Here, we show you a step-by-step solved example of linear differential equation. This solution was automatically generated by our smart calculator:

$x\frac{dy}{dx}-2y=x^3cos\left(x\right)$
2

Divide all the terms of the differential equation by $x$

$\frac{x}{x}\frac{dy}{dx}+\frac{-2y}{x}=\frac{x^3\cos\left(x\right)}{x}$

Simplify the fraction $\frac{x}{x}$ by $x$

$1\left(\frac{dy}{dx}\right)+\frac{-2y}{x}=\frac{x^3\cos\left(x\right)}{x}$

Any expression multiplied by $1$ is equal to itself

$\frac{dy}{dx}+\frac{-2y}{x}=\frac{x^3\cos\left(x\right)}{x}$

Simplify the fraction $\frac{x^3\cos\left(x\right)}{x}$ by $x$

$\frac{dy}{dx}+\frac{-2y}{x}=x^{2}\cos\left(x\right)$
3

Simplifying

$\frac{dy}{dx}+\frac{-2y}{x}=x^{2}\cos\left(x\right)$
4

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 $P(x)=\frac{-2}{x}$ and $Q(x)=x^{2}\cos\left(x\right)$. 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

$\int\frac{-2}{x}dx$

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

$-2\ln\left|x\right|$
5

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

$\int P(x)dx=\int\frac{-2}{x}dx=-2\ln\left(x\right)$

Simplify $e^{-2\ln\left|x\right|}$ by applying the properties of exponents and logarithms

$x^{-2}$
6

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

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

When multiplying exponents with same base we can add the exponents

$\frac{dy}{dx}x^{-2}+\frac{-2y}{x}x^{-2}=x^{2-2}\cos\left(x\right)$

Add the values $2$ and $-2$

$\frac{dy}{dx}x^{-2}+\frac{-2y}{x}x^{-2}=x^{0}\cos\left(x\right)$

Any expression (except $0$ and $\infty$) to the power of $0$ is equal to $1$

$\frac{dy}{dx}x^{-2}+\frac{-2y}{x}x^{-2}=\cos\left(x\right)$

Multiplying the fraction by $x^{-2}$

$\frac{dy}{dx}x^{-2}+\frac{-2yx^{-2}}{x}=\cos\left(x\right)$

Simplify the fraction $\frac{-2yx^{-2}}{x}$ by $x$

$\frac{dy}{dx}x^{-2}-2yx^{-3}=\cos\left(x\right)$
7

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

$\frac{dy}{dx}x^{-2}-2yx^{-3}=\cos\left(x\right)$
8

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

$\frac{d}{dx}\left(x^{-2}y\right)=\cos\left(x\right)$
9

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

$\int\frac{d}{dx}\left(x^{-2}y\right)dx=\int\cos\left(x\right)dx$
10

Simplify the left side of the differential equation

$x^{-2}y=\int\cos\left(x\right)dx$

Apply the integral of the cosine function: $\int\cos(x)dx=\sin(x)$

$\sin\left(x\right)$

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

$\sin\left(x\right)+C_0$
11

Solve the integral $\int\cos\left(x\right)dx$ and replace the result in the differential equation

$x^{-2}y=\sin\left(x\right)+C_0$

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\frac{1}{x^{\left|-2\right|}}y$

Multiplying the fraction by $y$

$\frac{y}{x^{\left|-2\right|}}$
12

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\frac{1}{x^{2}}y=\sin\left(x\right)+C_0$

Multiply the fraction by the term $y$

$\frac{1y}{x^{2}}=\sin\left(x\right)+C_0$

Any expression multiplied by $1$ is equal to itself

$\frac{y}{x^{2}}=\sin\left(x\right)+C_0$
13

Multiply the fraction by the term $y$

$\frac{y}{x^{2}}=\sin\left(x\right)+C_0$

Multiply both sides of the equation by $x^{2}$

$y=x^{2}\left(\sin\left(x\right)+C_0\right)$
14

Find the explicit solution to the differential equation. We need to isolate the variable $y$

$y=x^{2}\left(\sin\left(x\right)+C_0\right)$

Final answer to the problem

$y=x^{2}\left(\sin\left(x\right)+C_0\right)$

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