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Linear Differential Equation Calculator

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Example

Solved Problems

Difficult Problems

1

Solved example of linear differential equation

$x\frac{dy}{dx}-4y=x^6e^x$
2

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

$\frac{x}{x}\frac{dy}{dx}+\frac{-4y}{x}=\frac{x^6e^x}{x}$

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

$1\left(\frac{dy}{dx}\right)$

Any expression multiplied by $1$ is equal to itself

$\frac{dy}{dx}$

Any expression multiplied by $1$ is equal to itself

$\frac{dy}{dx}+\frac{-4y}{x}=\frac{x^6e^x}{x}$

Simplify the fraction $\frac{x^6e^x}{x}$ by $x$

$\frac{dy}{dx}+\frac{-4y}{x}=x^{5}e^x$
3

Simplifying

$\frac{dy}{dx}+\frac{-4y}{x}=x^{5}e^x$

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 $P(x)=\frac{-4}{x}$ and $Q(x)=x^{5}e^x$. 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

$\int\frac{-4}{x}dx$

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

$-4\ln\left(x\right)$
5

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

$\int P(x)dx=\int\frac{-4}{x}dx=-4\ln\left(x\right)$

Simplify $e^{-4\ln\left(x\right)}$ by applying the properties of exponents and logarithms

$x^{-4}$
6

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

$\mu(x)=x^{-4}$

When multiplying exponents with same base we can add the exponents

$\frac{dy}{dx}x^{-4}+\frac{-4y}{x}x^{-4}=xe^x$

Multiplying the fraction by $x^{-4}$

$\frac{dy}{dx}x^{-4}+\frac{-4yx^{-4}}{x}=xe^x$

Simplify the fraction $\frac{-4yx^{-4}}{x}$ by $x$

$\frac{dy}{dx}x^{-4}-4yx^{-5}=xe^x$
7

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

$\frac{dy}{dx}x^{-4}-4yx^{-5}=xe^x$
8

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

$\frac{d}{dx}\left(x^{-4}y\right)=xe^x$
9

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

$\int\frac{d}{dx}\left(x^{-4}y\right)dx=\int xe^xdx$
10

Simplify the left side of the differential equation

$x^{-4}y=\int xe^xdx$

We can solve the integral $\int xe^xdx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=x}\\ \displaystyle{du=dx}\end{matrix}$

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=e^xdx}\\ \displaystyle{\int dv=\int e^xdx}\end{matrix}$

Solve the integral

$v=\int e^xdx$

The integral of the exponential function is given by the following formula $\displaystyle \int a^xdx=\frac{a^x}{\ln(a)}$, where $a > 0$ and $a \neq 1$

$e^x$

Now replace the values of $u$, $du$ and $v$ in the last formula

$e^x\cdot x-\int e^xdx$

The integral of the exponential function is given by the following formula $\displaystyle \int a^xdx=\frac{a^x}{\ln(a)}$, where $a > 0$ and $a \neq 1$

$e^x\cdot x-e^x$

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

$e^x\cdot x-e^x+C_0$
11

Solve the integral $\int xe^xdx$ and replace the result in the differential equation

$x^{-4}y=e^x\cdot x-e^x+C_0$

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

$\frac{1}{x^{\left|-4\right|}}y$

Multiplying the fraction by $y$

$\frac{y}{x^{\left|-4\right|}}$
12

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

$\frac{1}{x^{4}}y=e^x\cdot x-e^x+C_0$
13

Multiply the fraction and term

$\frac{y}{x^{4}}=e^x\cdot x-e^x+C_0$

Multiply both sides of the equation by $x^{4}$

$y=\left(e^x\cdot x-e^x+C_0\right)x^{4}$
14

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

$y=\left(e^x\cdot x-e^x+C_0\right)x^{4}$

Final Answer

$y=\left(e^x\cdot x-e^x+C_0\right)x^{4}$

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