Question

The solution of $\dfrac{{dy}}{{dx}} + y = {e^x}$
$
  a)\,2y = {e^{2x}} + c \\
  b)\,2y{e^x} = {e^x} + c \\
  c)\,2y{e^x} = {e^{2x}} + c \\
  d)\,2y{e^{2x}} = 2{e^x} + c \\
 $

Answer 1

Hint: In this type of question you have to find out the (IF) that is integrating factor and then use integration to solve this problem.

Complete step-by-step answer:
So question is $\dfrac{{dy}}{{dx}} + y = {e^x}$
Here $\dfrac{{dy}}{{dx}}$is the derivative of $y$ with respect to $x$
So we know if the equation is $\dfrac{{dy}}{{dx}} + f(x)y = g(x)$
Then Integrating Factor (IF) $ = {e^{\int {f(x)dx} }}$
And then the general solution is given by
$(IF)y = \int {(IF)g(x)dx} $
Now, let us find the (IF) of given equation $\dfrac{{dy}}{{dx}} + y = {e^x}$
Here $f(x) = 1\,\,\,\,\,\& \,\,\,\,g(x) = {e^x}$
(IF) $ = {e^{\int {f(x)dx} }}$
$ = {e^{\int {dx} }}$
And we know $\int {dx = x} $
So we get (IF) $ = {e^x}$
Now we know the general solution is given by
$(IF)y = \int {(IF)g(x)dx} $
Here,
$
  {e^x}y = \int {{e^x}{e^x}dx} \\
  {e^x}y = \int {{e^{2x}}dx} \\
 $
And we know $\int {{e^{ax}}dx = \dfrac{{{e^{ax}}}}{a}} + c$ , using this we get
${e^x}y = \dfrac{{{e^{2x}}}}{2} + {c^,}$
$
  2{e^x}y = {e^{2x}} + 2{c^,} \\
  2{e^x}y = {e^{2x}} + {c^{,,}} \\
 $

So, the correct answer is “Option C”.

Note: This type of differential equation is solved by finding the integration factor and putting into the formula for equation of differential equation, if differential equation is given by $\dfrac{{dy}}{{dx}} + f(x)y = g(x)$ then (IF) $ = {e^{\int {f(x)dx} }}$and general solution is given by $(IF)y = \int {(IF)g(x)dx} $