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Exercise

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

Step-by-step Solution

1

Divide all the terms of the differential equation by $x-2$

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

Simplifying

$\frac{dy}{dx}+\frac{y}{x-2}=x+2$
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 $P(x)=\frac{1}{x-2}$ and $Q(x)=x+2$. In order to solve the differential equation, the first step is to find the integrating factor $\mu(x)$

$\displaystyle\mu\left(x\right)=e^{\int P(x)dx}$
4

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

$\int P(x)dx=\int\frac{1}{x-2}dx=\ln\left(x-2\right)$
5

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

$\mu(x)=x-2$
6

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

$\frac{dy}{dx}\left(x-2\right)+\frac{y}{x-2}\left(x-2\right)=\left(x+2\right)\left(x-2\right)$
7

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

$\frac{d}{dx}\left(\left(x-2\right)y\right)=\left(x+2\right)\left(x-2\right)$
8

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

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

Simplify the left side of the differential equation

$\left(x-2\right)y=\int\left(x+2\right)\left(x-2\right)dx$
10

Solve the product of difference of squares $\left(x+2\right)\left(x-2\right)$

$\left(x-2\right)y=\int\left(x^2-4\right)dx$
11

Expand the integral $\int\left(x^2-4\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

$\left(x-2\right)y=\int x^2dx+\int-4dx$
12

Solve the integral $\int x^2dx+\int-4dx$ and replace the result in the differential equation

$\left(x-2\right)y=\frac{x^{3}}{3}-4x+C_0$
13

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

$y=\frac{x^{3}-12x+C_1}{3\left(x-2\right)}$

Final answer to the exercise

$y=\frac{x^{3}-12x+C_1}{3\left(x-2\right)}$

Try other ways to solve this exercise

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  • Exact Differential Equation
  • Linear Differential Equation
  • Separable Differential Equations
  • Homogeneous Differential Equation
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Integrate by substitution
  • Integrate by parts
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