# Thread: Second Order Differential Equation

1. ## Second Order Differential Equation

how to solve the second order differential equation in the form:
ay" + by' + cy = R(x)?

the example is this:
(d2i/dt2) + 6(di/dt) +25i = -292sin(4t)

First you find the general solution of the corresponding homogeneous equations ie. $ay'' + by' + cy = 0$.
Then you look for a particular solution, since in this case $g(t) = -292\sin{4t}$ we can guess that the particular solution will be of the form $Y(t) = A\cos{4t} + B\sin{4t}$. Now you solve for $A$ and $B$ obviously, by substituting in the appropriate terms ie. $y''$ and $y'$ and $y$ into your original differential equation. Then you know the general form of the solution of nonhomogeneous equation is simply $y = \phi(t) = c_1y_1(t) + c_2y_2(t) + Y(t)$, where $y_1(t)$ and $y_2(t)$ are solutions of the homogeneous equation and $Y(t)$ is the particular solution we guessed and solved the constants $A$ and $B$ for. And $c_1$ and $c_2$ are arbitrary constants.