For dy/dx+ay= x, with constants a,ω and y(0)=1, select true statements as x.

(C) $y\to A\exp(|a|x)$ if $a 0$, B,C constants

GATE 2023 AE – Question 42

Engineering Mathematics · Differential Equations: First order linear ODEs and higher order linear ODEs with constant coefficients · 2 marks · Multiple select

For $dy/dx+ay=\sin\omega x$, with constants a,ω and y(0)=1, select true statements as $x\to\infty$.

$y\to0$ if $a\ne0$
$y\to1$ if $a=0$
$y\to A\exp(|a|x)$ if $a<0$, A constant
$y\to B\sin(\omega x+C)$ if $a>0$, B,C constants

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Correct answer: (C) $y\to A\exp(|a|x)$ if $a<0$, A constant; (D) $y\to B\sin(\omega x+C)$ if $a>0$, B,C constants

Explanation

**Solve the first-order linear equation** $y^\prime+ay=\sin\omega x$ with an integrating factor $e^{ax}$.

The general solution is
$$y=Ce^{-ax}+y_p(x),$$
where, for $a\neq0$, the particular solution is sinusoidal with the same frequency:
$$y_p=\frac{a\sin\omega x-\omega\cos\omega x}{a^2+\omega^2}=B\sin(\omega x+C_1).$$

**Check each statement.**
- **A. $y\to0$ if $a\neq0$.** False. For $a>0$ the transient dies away but the sinusoidal part stays, so $y$ does not go to 0. For $a<0$ it blows up.
- **B. $y\to1$ if $a=0$.** False. With $a=0$, $y=1+\dfrac{1-\cos\omega x}{\omega}$, which oscillates.
- **C. $y\to A\exp(|a|x)$ if $a<0$.** True. When $a<0$, $e^{-ax}=e^{|a|x}$ grows and dominates. ✓
- **D. $y\to B\sin(\omega x+C)$ if $a>0$.** True. The term $Ce^{-ax}$ decays and only the sinusoid remains. ✓

Answer **C and D**.

Official GATE 2023 answer key: https://gate2026.iitg.ac.in/doc/download/Answer_keys2023/AE_ANS_GATE2023.pdf