Differential Equations Question 143
Question: Solution of differential equation $ 2xy\frac{dy}{dx}=x^{2}+3y^{2} $ is
[MP PET 1993]
Options:
A) $ x^{3}+y^{2}=px^{2} $
B) $ \frac{x^{2}}{2}+\frac{y^{3}}{x}=y^{2}+p $
C) $ x^{2}+y^{3}=px^{2} $
D) $ x^{2}+y^{2}=px^{3} $
Show Answer
Answer:
Correct Answer: D
Solution:
It is homogeneous equation $ \frac{dy}{dx}=\frac{x^{2}+3y^{2}}{2xy} $
Put $ y=vx $ and $ \frac{dy}{dx}=v+x\frac{dv}{dx} $
So, we get $ x\frac{dv}{dx}=\frac{1+v^{2}}{2v} $
Therefore $ \frac{2vdv}{1+v^{2}}=\frac{dx}{x} $
On integrating, we get $ x^{2}+y^{2}=px^{3} $ . (where p is a constant)
BETA
