Here, we show you a step-by-step solved example of linear differential equation. This solution was automatically generated by our smart calculator:
Divide all the terms of the differential equation by $x$
Simplify the fraction
Any expression multiplied by $1$ is equal to itself
Any expression multiplied by $1$ is equal to itself
Simplify the fraction $\frac{x^6e^x}{x}$ by $x$
Simplifying
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 $\mu(x)$
Compute the integral
The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$
To find $\mu(x)$, we first need to calculate $\int P(x)dx$
Simplify $e^{-4\ln\left|x\right|}$ by applying the properties of exponents and logarithms
So the integrating factor $\mu(x)$ is
When multiplying exponents with same base we can add the exponents
Multiplying the fraction by $x^{-4}$
Simplify the fraction $\frac{-4yx^{-4}}{x}$ by $x$
Now, multiply all the terms in the differential equation by the integrating factor $\mu(x)$ and check if we can simplify
We can recognize that the left side of the differential equation consists of the derivative of the product of $\mu(x)\cdot y(x)$
Integrate both sides of the differential equation with respect to $dx$
Simplify the left side of the differential equation
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
First, identify or choose $u$ and calculate it's derivative, $du$
Now, identify $dv$ and calculate $v$
Solve the integral to find $v$
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 $a \neq 1$
Now replace the values of $u$, $du$ and $v$ in the last formula
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 $a \neq 1$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Solve the integral $\int xe^xdx$ and replace the result in the differential equation
Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number
Multiplying the fraction by $y$
Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number
Multiply the fraction and term
Multiply both sides of the equation by $x^{4}$
Find the explicit solution to the differential equation. We need to isolate the variable $y$
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