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Exercise

x3dydx+3x2y=xx^3\frac{dy}{dx}+3x^2y=x

Step-by-step Solution

1

Divide all the terms of the differential equation by x3x^3

x3x3dydx+3x2yx3=xx3\frac{x^3}{x^3}\frac{dy}{dx}+\frac{3x^2y}{x^3}=\frac{x}{x^3}
2

Simplifying

dydx+3yx=1x2\frac{dy}{dx}+\frac{3y}{x}=\frac{1}{x^{2}}
3

We can identify that the differential equation has the form: dydx+P(x)⋅y(x)=Q(x)\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)=3xP(x)=\frac{3}{x} and Q(x)=1x2Q(x)=\frac{1}{x^{2}}. In order to solve the differential equation, the first step is to find the integrating factor μ(x)\mu(x)

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

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

∫P(x)dx=∫3xdx=3ln⁡(x)\int P(x)dx=\int\frac{3}{x}dx=3\ln\left(x\right)
5

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

μ(x)=x3\mu(x)=x^3
6

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

dydxx3+3yx2=x\frac{dy}{dx}x^3+3yx^{2}=x
7

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

ddx(x3y)=x\frac{d}{dx}\left(x^3y\right)=x
8

Integrate both sides of the differential equation with respect to dxdx

∫ddx(x3y)dx=∫xdx\int\frac{d}{dx}\left(x^3y\right)dx=\int xdx
9

Simplify the left side of the differential equation

x3y=∫xdxx^3y=\int xdx
10

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

x3y=12x2+C0x^3y=\frac{1}{2}x^2+C_0
11

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

y=x2+C12x3y=\frac{x^2+C_1}{2x^3}

Final answer to the exercise

y=x2+C12x3y=\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 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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x3dydx +3x2y=x
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