Differential Equations Question 45
Question: The equation of the curve passing through the origin and satisfying the equation $ (1+x^{2})\frac{dy}{dx}+2xy=4x^{2} $ is
Options:
A) $ 3(1+x^{2})y=4x^{3} $
B) $ 3(1-x^{2})y=4x^{3} $
C) $ 3(1+x^{2})=x^{3} $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
$ \frac{dy}{dx}+\frac{2x}{1+x^{2}}y=\frac{4x^{2}}{1+x^{2}} $
It is linear equation of the form $ \frac{dy}{dx}+Py=Q $
Here $ P=\frac{2x}{1+x^{2}} $ and $ Q=\frac{4x^{2}}{1+x^{2}} $
I.F. $ ={e^{\int _{{}}^{{}}{\frac{2x}{1+x^{2}}dx}}}={e^{\log (1+x^{2})}}=(1+x^{2}) $
Therefore, solution is given by $ y.(1+x^{2})=\int{\frac{4x^{2}}{1+x^{2}}(1+x^{2})dx+c=\frac{4x^{3}}{3}}+c $ . But it passes through (0,0) therefore $ c=0 $ , hence the curve is $ 3y(1+x^{2})=4x^{3} $ .
BETA
