How to solve 2nd order diff. eqn.

[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.

 

[paranormal]

The method for solving a second order differential equation, such as the one provided, is known as the "method of undetermined coefficients." This method involves assuming a solution for the equation in the form of a polynomial or exponential function and then substituting it into the equation to solve for the unknown coefficients. The specific steps for solving the equation would depend on the initial conditions and boundary conditions given. Another method that can be used is the "method of variation of parameters," which involves finding a particular solution by varying the parameters in the assumed solution. Both of these methods require a good understanding of calculus and algebra, as well as knowledge of specific techniques for solving differential equations. It is also important to check the solution obtained to ensure it satisfies the original equation.