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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 additional solution(s) of the given differential equation is(are)
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$y=5 u(x)+8 v(x)$, $y=c_{1}\{u(x)-v(x)\}+c_{2} v(x), c_{1}$ and $c_{2}$ are arbitrary constants
We know that $u(x)$ and $v(x)$ are two independent solutions of the given differential equation, then their linear combination is also the solution of the given equation.
Here, we see that $y=5 u(x)+8 v(x)$ is a linear combination and $y=c_{1}\{u(x)-v(x)\}+c_{2} v(x)$ is also a linear combination of two independent solutions.
Here, we see that $y=5 u(x)+8 v(x)$ is a linear combination and $y=c_{1}\{u(x)-v(x)\}+c_{2} v(x)$ is also a linear combination of two independent solutions.
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