Solving First Order ODE with Integrating Factor

[bobred]

Homework Statement

Solve first order ODE

Homework Equations

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

The Attempt at a Solution

Integrating factor
$$p=\exp(-\int \frac{2}{x})=\exp(-2\ln x)=x^{-2}$$

Multiplying through by the integrating factor
$$\frac{d}{dy}(x^{-2}y)=x^{-2}$$

Integrating both sides
$$x^{-2}y=-x^{-1}+C$$

Dividing through by $$x^{-2}$$
$$y=Cx^2-x$$

The problem comes when I use say, Maple to check the answer, it gives

$$y=x^3+Cx^2-x$$

Any ideas? Thanks

 

[HallsofIvy]

Homework Statement

Solve first order ODE

Homework Equations

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

The Attempt at a Solution

Integrating factor
$$p=\exp(-\int \frac{2}{x})=\exp(-2\ln x)=x^{-2}$$

Multiplying through by the integrating factor
$$\frac{d}{dy}(x^{-2}y)=x^{-2}$$

No, the right hand side of your original equation was $x^2+ 1$. Multiplying that by $x^{-2}$ gives $1+ x^{-2}$. You've dropped the "1".

Integrating both sides
$$x^{-2}y=-x^{-1}+C$$

Dividing through by $$x^{-2}$$
$$y=Cx^2-x$$

The problem comes when I use say, Maple to check the answer, it gives

$$y=x^3+Cx^2-x$$

Any ideas? Thanks

 

[bobred]

Thanks