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Question

(A) The given differential equation $\frac{d^{2} y}{d x^{2}}+b \frac{d y}{d x}+c y=0$ is a linear homogeneous differential equation of the second orde

Vedclass · Accepted Answer

$y=5 u(x)+8 v(x)$

Vedclass · Answer

(A) The given differential equation $\frac{d^{2} y}{d x^{2}}+b \frac{d y}{d x}+c y=0$ is a linear homogeneous differential equation of the second order. By the principle of superposition, if $u(x)$ and $v(x)$ are two independent solutions, then any linear combination $y = c_{1}u(x) + c_{2}v(x)$ is also a solution, where $c_{1}$ and $c_{2}$ are arbitrary constants. Option $A$ is a specific case of a linear combination where $c_{1}=5$ and $c_{2}=8$. Option $B$ can be rewritten as $y = c_{1}u(x) + (c_{2}-c_{1})v(x)$, which is also a linear combination of $u(x)$ and $v(x)$ with new arbitrary constants. Therefore, both $A$ and $B$ represent solutions to the differential equation.

If $u(x)$ and $v(x)$ are two independent solutions of the differential equation $\frac{d^{2} y}{d x^{2}}+b \frac{d y}{d x}+c y=0$, then which of the following is also a solution of the given differential equation?

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