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Exercise

$\frac{dy}{dx}+\frac{3}{x}y=\frac{1}{x^2}$

Step-by-step Solution

1

Multiplying the fraction by $y$

$\frac{dy}{dx}+\frac{3y}{x}=\frac{1}{x^2}$
2

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 $P(x)=\frac{3}{x}$ and $Q(x)=\frac{1}{x^2}$. In order to solve the differential equation, the first step is to find the integrating factor $\mu(x)$

$\displaystyle\mu\left(x\right)=e^{\int P(x)dx}$
3

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

$\int P(x)dx=\int\frac{3}{x}dx=3\ln\left(x\right)$
4

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

$\mu(x)=x^3$
5

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

$\frac{dy}{dx}x^3+3yx^{2}=x$
6

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

$\frac{d}{dx}\left(x^3y\right)=x$
7

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

$\int\frac{d}{dx}\left(x^3y\right)dx=\int xdx$
8

Simplify the left side of the differential equation

$x^3y=\int xdx$
9

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

$x^3y=\frac{1}{2}x^2+C_0$
10

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

$y=\frac{x^2+C_1}{2x^3}$

Final answer to the exercise

$y=\frac{x^2+C_1}{2x^3}$

Try other ways to solve this exercise

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  • Exact Differential Equation
  • Linear Differential Equation
  • Separable Differential Equation
  • 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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Main Topic: Integrals of Polynomial Functions

Integrals of polynomial functions.

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