Choose the correct answer from the given four options.The differe… - Vidyadip

MCQ

Choose the correct answer from the given four options.The differential equation for which $\text{y}=\text{a}\cos\text{x}+\text{b}\sin\text{x}$ is a solution, is:

✓
$\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}+\text{y}=0$
B
$\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}-\text{y}=0$
C
$\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}+(\text{a}+\text{b})\text{y}=0$
D
$\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}+(\text{a}-\text{b})\text{y}=0$

✓

Answer

Correct option: A.

$\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}+\text{y}=0$

Given equation is, $\text{y}=\text{a}\cos\text{x}+\text{b}\sin\text{x}$
On differentiating both sides $\text{w.r.t.x.}$ we get
$\frac{\text{dy}}{\text{dx}}=-\text{a}\sin\text{x}+\text{b}\cos\text{dx}$
Again, differentiating $\text{w.r.t.x.}$ we get
$\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}=-\text{a}\sin\text{x}+\text{b}\cos\text{dx}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}=-\text{y}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}+\text{y}=0$

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