How to Solve a Second Order Differential Equation?

[physiker99]

what's the method to solve a diff. equation as follows:

d^2(psi)/d(x^2) - (k^2)*(psi) = 0

 

[kreil]

Notice that:

$$\frac{d^2 \psi}{dx^2}=k^2 \psi$$

And the general solution is,

$$\psi(x) = Ae^{kx} + Be^{-kx}$$

Since

$$\frac{d^2 \psi}{dx^2} = k^2(Ae^{kx} + Be^{-kx}) = k^2 \psi(x)$$

 

[physiker99]

thanks kreil. but how do you get a*e^kx and b*e^-kx?

 

[vela]

When you have a linear differential equation with constant coefficients, like yours, you assume a solution of the form y=e^rx and substitute it into the equation. You'll get what's called a characteristic polynomial. Its roots are the values of r for which your assumed solution will satisfy the differential equation.