Differential-Equations Question 378
Question: The solution of the differential equation $ {{(x+y)}^{2}}\frac{dy}{dx}=a^{2} $ is
[AMU 2001]
Options:
A) $ {{(x+y)}^{2}}=\frac{a^{2}}{2}x+c $
B) $ {{(x+y)}^{2}}=a^{2}x^{2}+c $
C) $ {{(x+y)}^{2}}=2a^{2}x+c $
D) None of these
Show Answer
Answer:
Correct Answer: D
Solution:
Put $ x+y=v $
Þ $ 1+\frac{dy}{dx}=\frac{dv}{dx} $
Þ $ \frac{dy}{dx}=\frac{dv}{dx}-1 $ \ $ v^{2}( \frac{dv}{dx}-1 )=a^{2} $
Þ $ \frac{dv}{dx}=\frac{a^{2}}{v^{2}}+1=\frac{a^{2}+v^{2}}{v^{2}} $
Þ $ \frac{v^{2}}{a^{2}+v^{2}}dv=dx $
Þ $ ( 1-\frac{a^{2}}{a^{2}+v^{2}} )dv=dx $
Þ $ v - a\tan^{-1}\left(\frac{v}{a}\right) = x + c $
Þ $ y=a{{\tan }^{-1}}( \frac{x+y}{a} ) $ + c.
BETA
