Differential Equations Question 170
Question: The solution of the differential equation $ x\frac{d^{2}y}{dx^{2}}=1 $ , given that $ y=1,\ \frac{dy}{dx}=0 $ when $ x=1 $ , is
Options:
A) $ y=x\log x+x+2 $
B) $ y=x\log x-x+2 $
C) $ y=x\log x+x $
D) $ y=x\log x-x $
Show Answer
Answer:
Correct Answer: B
Solution:
$ x\frac{d^{2}y}{dx^{2}}=1 $
Therefore $ \frac{d^{2}y}{dx^{2}}=\frac{1}{x} $
Therefore $ \frac{dy}{dx}=\log x+c_1 $
Therefore $ y=x\log x-x+c_1x+c_2 $
(on integrating twice) Given $ y=1 $ and $ \frac{dy}{dx}=0 $ at $ x=1 $
Therefore $ c_1=0 $ and $ c_2=2 $ Therefore, the required solution is $ y=x\log x-x+2 $ .
BETA
