The differential equation whose solution is y = c_1 ax + c_2 ax i… - Vidyadip

MCQ

The differential equation whose solution is $y = {c_1}\cos ax + {c_2}\sin ax$ is(Where ${c_1},\;{c_2}$are arbitrary constants)

A
$\frac{{{d^2}y}}{{d{x^2}}} + {y^2} = 0$
✓
$\frac{{{d^2}y}}{{d{x^2}}} + {a^2}y = 0$
C
$\frac{{{d^2}y}}{{d{x^2}}} + a{y^2} = 0$
D
$\frac{{{d^2}y}}{{d{x^2}}} - {a^2}y = 0$

✓

Answer

Correct option: B.

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

b
(b) Solution is $y = {c_1}\cos ax + {c_2}\sin ax$

Differentiate it w.r.t. $x$, we get

$\frac{{dy}}{{dx}} = - {c_1}a\sin ax + {c_2}a\cos ax$

Again $\frac{{{d^2}y}}{{d{x^2}}} = - {c_1}{a^2}\cos ax - {c_2}{a^2}\sin ax$

$\frac{{{d^2}y}}{{d{x^2}}} = - {a^2}({c_1}\cos ax + {c_2}\sin ax) \Rightarrow \frac{{{d^2}y}}{{d{x^2}}} = - {a^2}y$

or $\frac{{{d^2}y}}{{d{x^2}}} + {a^2}y = 0$.

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