## Second Order Differential equations

**Introduction: **The simplest equations are constant coefficients such:

Substituting e^{s}t results in the quadratic equation

If b^{2 }> 4 a c then there are two unique solutions for the quadratic equation and the general solution is

For the case of b^{2}= 4 a c the general solution is

In the case of b^{2}> 4 a c the solution of the quadratic equation is a complex number which means that the solution has exponential and trigonometric functions as

Where the real part is

And the imaginary number is

**Non-Homogeneous Equation:**

Equation that not equal to zero in this form

The solution of the homogeneous equation is zero so additional solution of l(x) though the operation the right hand side is the total solution as

Where the solution uh is the solution of the homogeneous solution and up is the solution of the particular function l(x). If the function on the right hand side is polynomial than the solution is will

**Non-Linear Second Order Equation:**

Some of the techniques that were discussed in the previous can be used for the second order differential equation such variable separation. If the following equation

Can be written in the form

Then the equation it is referred that equation is separable. The derivative of u˙ can be treated as new function v and v˙ = u¨. Hence, equation can be integrated

The integration results in a first order differential equation which should be dealt with the previous methods. It can be noticed that initial condition of the function is used twice. The physical reason is that the equation represents strong effect of the function at a certain point.