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