# Solve $(x+y)^2 dy=a^2 dx$ as an Ordinary Linear Differential Equation

Found this problem in Advanced Engineering Mathematics By H.K. Dass Under "Ordinary Linear Differential Equation" (page 165 Question 19).

$$(x+y)^2 \frac{dy}{dx} = a^2$$

I can figure it out how to solve this as an separable variable D.E. by substituting $x+y=z$. But since this was under Linear D.E. section, I think there should be a way to solve this as an O.L.D.E.

Any help would be appreciated.

The Answer Is

$$y+x = a \tan(\frac{y-c}{a})$$

What I'm Looking For is

To solve this as an Linear Differential Equation. i.e.

$$\frac{dy}{dx} + P(x) y = Q(x)$$

• Yeah I got the answer in that approach. But what i want is to solve it as a Linear differential equation. – Gayan Kavirathne Jan 2 '17 at 7:36
• use $y+x=u$ and it simplifies a lot :-) – Math-fun Jan 2 '17 at 7:59
• @Moo Can you describe it further.. please – Gayan Kavirathne Jan 2 '17 at 8:27

## 1 Answer

Your equation $(x+y)^2\frac{dy}{dx}=a^2$ is a non-linear differential equation and your approach looks fine to me. However, on substituting $x+y=z$, the reduced form

$\frac{dz}{dx}=\frac{z^2+a^2}{z^2}$ is still non-linear, so the author has wrongly incorporated this particular example in the exercise.

• Thanks.. But i'll leave this question opened in case someone would come up with an answer. Thanks a lot @mathlover – Gayan Kavirathne Jan 2 '17 at 8:50
• My pleasure...As I stated the approach you are looking for is not possible as the given ODE is non-linear. The book you are following has lots of errors so be careful. – Mathlover Jan 2 '17 at 10:46
• Thanks.. I think you are correct. I found some other mistakes too. Can you recommend me some better book to refer.! Thanks again – Gayan Kavirathne Jan 2 '17 at 16:10
• Engineering Mathematics by Ravish kumar (Mcgrawhill Publication India) – Mathlover Jan 2 '17 at 17:31