Question

The differential equation whose solution is $y = ax + b{e^x}:$
a. $(x - 1){y_2} - x{y_1} + y = 0$
b. $(x - 1){y_2} - x{y_1} = y$
c. ${x^2}{y_2} - x{y_1} + y = 0$
d. ${x^2}{y_2} + x{y_1} - y = 0$

Answer 1

We can solve this type of differential equations by differentiating only. We first differentiate the equation with respect to x to eliminate the x from the first term. And then we differentiate that equation again with respect to x, this will give the value of constant b. Putting the values of a and b in the first equation will give us our needed result.

Complete step-by-step answer:
$y = ax + b{e^x}$…….(i)
Differentiating with respect to x we get,
\Rightarrow $$$${y_1} = a + b{e^x} where, ${y_1} = \dfrac{{dy}}{{dx}}$ ……….(ii)
Again, Differentiating with respect to x we get,
$\Rightarrow$ ${y_2} = b{e^x}$ where, ${y_2} = \dfrac{{{d^2}y}}{{d{x^2}}}$
Then,
$\Rightarrow$ $b = {y_2}{e^{ - x}}$
Substituting the value of b in equation (ii), we get,
$\Rightarrow$ ${y_1} = a + {y_2}{e^{ - x}}.{e^x}$
On simplification we get,
$\Rightarrow {y_1} = a + {y_2}$
$\Rightarrow a = {y_1} - {y_2}$
Again, putting the value of a and b in equation (i), we get,
$y = ({y_1} - {y_2})x + {y_2}{e^{ - x}}.{e^x}$
On simplification we get,
$\Rightarrow y = x{y_1} - x{y_2} + {y_2}$
On taking ${y_2}$ common from last two terms we get,
$\Rightarrow y = x{y_1} - {y_2}(x - 1)$
On rearranging we get,
$\Rightarrow (x - 1){y_2} - x{y_1} + y = 0$
So, we have our answer as option a, $(x - 1){y_2} - x{y_1} + y = 0$

Note: In this type of problems the target of the problem is to find the values of distinct constants a and b. Then putting the values of a and b into our given equation will give us our desired result.
Basically we can say that we are eliminating the constants to get a differential equation.