DIFFERENTIAL EQUATION_JEE Advanced_2021_S2 - Competishun – Live JEE & NEET Courses and Exam Prep

JEE Advanced 2021

Paper-2 2021

Multiple correct answers - Select all that apply

Question

For any real numbers $\alpha$ and $\beta$, let $y_{\alpha, \beta}(x), x \in \mathbb{R}$, be the solution of the differential equation
$ \frac{d y}{d x}+\alpha y=x e^{\beta x}, y(1)=1 $
Let $S=\left\{y_{\alpha, \beta}(x): \alpha, \beta \in \mathbb{R}\right\}$. Then which of the following functions belong(s) to the set $S$ ?

Select ALL correct options:

$f(x)=\frac{x^2}{2} e^{-x}+\left(e-\frac{1}{2}\right) e^{-x}$

$f(x)=-\frac{x^2}{2} e^{-x}+\left(e+\frac{1}{2}\right) e^{-x}$

$f(x)=\frac{e^x}{2}\left(x-\frac{1}{2}\right)+\left(e-\frac{e^2}{4}\right) e^{-x}$

$f(x)=\frac{e^x}{2}\left(\frac{1}{2}-x\right)+\left(e+\frac{e^2}{4}\right) e^{-x}$

✓ Correct! Well done.

✗ Incorrect. Try again or view the solution.

⚠ Partially correct. Some answers are missing.

Solution