Question

The differential equation of the family of curves $y = a{e^x} + bx{e^x} + c{x^2}{e^x}$, where $a,b$ and $c$ are arbitrary constants, is:
A. ${y^{\prime \prime \prime }} + 3y'' + 3y' + y = 0$
B. ${y^{\prime \prime \prime }} + 3y'' - 3y' - y = 0$
C. ${y^{\prime \prime \prime }} - 3y'' - 3y' + y = 0$
D. ${y^{\prime \prime \prime }} - 3y'' + 3y' - y = 0$

Answer 1

In order to solve this question we have to differentiate the family of curve $y = a{e^x} + bx{e^x} + c{x^2}{e^x}$ and there are three arbitrary constants in the equation and we have to differentiate this equation three times and eliminate all the arbitrary constant from all the three equation. And final equation in terms and x, and y.

Complete step by step answer:
We have given a family of curves and we have to find the differential equation of this curve.
In this $y = a{e^x} + bx{e^x} + c{x^2}{e^x}$ equation of curve we have given three arbitrary constants so we have to differentiate this equation three times.
$\dfrac{{dy}}{{dx}}$ is represented by $y'$.
$\Rightarrow y = a{e^x} + bx{e^x} + c{x^2}{e^x}$
On differentiating with respect to x,
$y' = a{e^x} + b\left( {x{e^x} + {e^x}} \right) + c\left( {{x^2}{e^x} + 2x{e^x}} \right)$

On simplifying this equation.
$y' = a{e^x} + bx{e^x} + b{e^x} + c{x^2}{e^x} + c2x{e^x}$
Now expressing this term in terms of y.
$y' = y + b{e^x} + c2x{e^x}$
Now, again differentiating with respect to x.
$y'' = y' + b{e^x} + 2c\left( {x{e^x} + {e^x}} \right)$
On further solving
$y'' = y' + b{e^x} + 2cx{e^x} + 2c{e^x}$

Now putting the value in terms of $y'$ and $y$ in this equation.
$y'' = 2y' - y + 2c{e^x}$
Again differentiating with respect to x.
$y''' = 2y'' - y' + 2c{e^x}$
Now putting the value of $2c{e^x}$ from the last obtained equation.
$y''' = 2y'' - y' + y'' - 2y' + y$
On further solving
$\therefore y''' - 3y'' + 3y' - y = 0$
Hence, the differential equation of family of curve $y = a{e^x} + bx{e^x} + c{x^2}{e^x}$ is $y''' - 3y'' + 3y' - y = 0$.

Therefore, option “D” is the correct answer.

Note: To solve these types of questions students must know the multiplication rule of the differentiation and students must know how they solve the equation. There are many places where students often make mistakes. To find a differential equation of a curve having n number of arbitrary constants then we have to differentiate that equation n times.