Differential Equations Question 348
Question: Under which one of the following conditions does the solution of $ \frac{dy}{dx}=\frac{ax+b}{cy+d} $ represent a parabola-
Options:
A) $ a=0,c=0 $
B) $ a=1,b=2,c\ne 0 $
C) $ a=0,c\ne 0,b\ne 0 $
D) $ a=1,c=1 $
Show Answer
Answer:
Correct Answer: C
Solution:
[c] Given: $ \frac{dy}{dx}=\frac{ax+b}{cy+d} $ or $ (cy+d)dy=(ax+b)dx $
Integrating both the sides. $ c.\int{ydy+d\int{dy=a\int{xdx+b\int{dx+K}}}} $ [K is constant integration] or, $ c.\frac{y^{2}}{2}+d.y=a\frac{x^{2}}{2}+b.x+K $ or, $ cy^{2}+2d.y=ax^{2}+2b.x+2K $
This equation will represent a parabola when either, the coefficient of $ x^{2} $ or the coefficient of $ y^{2} $ is zero, but not both.
Thus either c = 0 or a = 0 but not both. From the choice given, a = 0, $ c\ne 0 $ and $ b\ne 0 $ .
BETA
