Question

If $u\left(x\right)$ and $v\left(x\right)$ are two independent solutions of the differential equation
$\frac{d^{2}y}{d{x}^2}+b\frac{dy}{dx}+cy=0$, then additional solution(s) of the given differential equation is(are):

• A. $y=5u\left(x\right)+8v\left(x\right)$
• B. $y=c_1\{u\left(x\right)-v\left(x\right)\}+c_{2}v\left(x\right),c_1$ and $c_2$ are arbitrary constants
• C. $y=c_{1}u\left(x\right)v\left(x\right)+c_{2}u\left(x\right)/v\left(x\right),c_1$ and $c_2$ are arbitrary constants
• D. $y=u\left(x\right)v\left(x\right)$

Answer 1

Answer: (A), (B)

$y=5u\left(x\right)+8v\left(x\right)$
$y=c_1\{u\left(x\right)-v\left(x\right)\}+c_{2}v\left(x\right),c_1$ and $c_2$ are arbitrary constants

We know that $u\left(x\right)$ and $v\left(x\right)$ 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 $y=5u\left(x\right)+8v\left(x\right)$ is a linear combination and $y=c_1\{u\left(x\right)-v\left(x\right)\}+c_{2}v\left(x\right)$ is also a linear combination of two independent solutions.