Question

. Find the general solution to
d²y/dx²= 49y. Enter your answer as an equation y = ..

Note: d²y/dx²=k^2y

Answer 1

The given differential equation is linear with constant coefficients,

$\dfrac{d^2y}{dx^2} - 49y = 0$

with characteristic equation

$r^2 - 49 = 0$

and hence characteristic roots $r=\pm7$. This means the general solution to the ODE is

$y = C_1 e^{7x} + C_2 e^{-7x}$

In fact, you're given the solution already,

$y = C_1 e^{kx} + C_2 e^{-kx}$

and you've determined that

$\dfrac{d^2y}{dx^2} = k^2 (C_1 e^{kx} + C_2 e^{-kx}) = k^2 y$

Comparing this to the given ODE, it's obvious that $k=7$, so you can just replace $k$ with 7 in the given template solution.