Question

Solution of $\frac{dy}{dx}-y=1$, $y\left(0\right)=1$ is given by

• A. $xy=-e^x$
• B. $xy=-e^{-x}$
• C. $xy=-1$
• D. $y=2e^x-1$

Answer 1

Answer: (D)

$\frac{dy}{dx}-y=1$
$\Rightarrow \frac{dy}{dx}=1+y$
$\Rightarrow \frac{dy}{1+y}=dx$
On integrating, we get $log\left(1+y\right)=x+c$
Now, $y\left(0\right)=1$
$\Rightarrow log2=c$
$\therefore log\left(1+y\right)=x+log2$
$\Rightarrow \frac{1+y}{2}=e^x$
$\Rightarrow 1+y=2e^x$
$\Rightarrow y=2e^x-1$