Differential Equations Question 257
Question: The solution to of the differential equation $ (x+1)\frac{dy}{dx}-y=e^{3x}{{(x+1)}^{2}} $ is
Options:
A) $ y=(x+1)e^{3x}+c $
B) $ 3y=(x+1)+e^{3x}+c $
C) $ \frac{3y}{x+1}=e^{3x}+c $
D) $ y{e^{-3x}}=3(x+1)+c $
Show Answer
Answer:
Correct Answer: C
Solution:
[c] The given equation is $ \frac{dy}{dx}-\frac{y}{x+1}=e^{3x}(x+1) $ I.F. $ ={e^{\int{\frac{1}{x+1}dx}}}={e^{-\log (x+1)}}=\frac{1}{x+1} $ The solution is $ y( \frac{1}{x+1} )=\int{e^{3x}(x+1).\frac{1}{x+1}}dx+a $
$ \Rightarrow \frac{y}{x+1}=\int{e^{3x}dx+a=\frac{e^{3x}}{3}+a} $
$ \Rightarrow \frac{3y}{x+1}=e^{3x}+c,c=3a $
BETA
