The value of constants $m$ and $c$ for which $y = m x + c$ is a solution of the differential equation $D^{2} y + 3 D y + 4 y = 4 x$
$(D^{2} y = \frac{d^{2} y}{d x^{2}}, D y = \frac{d y}{d x})$

m = - 1

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c = - \frac{3}{4}

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c = \frac{3}{4}

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m = 1

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Solution

The correct option is D $m = 1$
Given : $y = m x + c$
Differentiating w.r.t. $x$
$\Rightarrow D y = m \dots (i)$
Again differentiating w.r.t $x$
$D^{2} y = 0 \dots (i i)$
Using $(i)$ and $(i i)$ in $D^{2} y + 3 D y + 4 y = 4 x,$ we get

$0 + 3 m + 4 (m x + c) = 4 x$
$\Rightarrow 3 m + 4 m x + 4 c = 4 x$
By comparing, we get
$m = 1$ and $c = - \frac{3}{4}$

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