Prove basic theorem on linear homogeneous differential equation for $m=n=2$ and if $f_1(x)$ and $f_2(x)$ are two solutions of

$$a_0(x)\frac{d^2y}{dx^2}+a_1(x)\frac{dy}{dx}+a_2(x)y=0$$ then $c_1f_1(x)$ and $c_2f_2(x)$ are also the solution to this equation where $c_1$ and $c_2$ are t arbitrary constants!

My attempt

Given that it is second order homogeneous linear differential equation we then have two solutions to the differential equation!

Knowing that $f_1(x)$ & $f_2(x)$ are solution to the differential equation Thus, we also express them as a linear combination of solutions to the differential equation!

That is by introducing $c_1$ and $c_2$. Thus, $c_1f_1(x) +c_2f_2(x)=0$ is also a solution!

I totally have no idea how to do proof. Can someone guide me?

  • 1
    $\begingroup$ 1. Write what is given carefully. 2. Understand what you need to show. Set $y(x)=c_1f_1(x)+c_2f_2(x)$. 3. Recall that derivative of a sum is the sum of the derivatives. 4. Conclude. $\endgroup$ – Artem Jun 19 '17 at 2:47
  • $\begingroup$ Meaning just replacing $y(x)=c_1f_1(x)+c_2f_2(x)$ in the the differential equation by differentiating w.r.t to the order? $\endgroup$ – Crazy Jun 19 '17 at 2:49

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