Second Order Differential equations
Introduction: The simplest equations are constant coefficients such:
Substituting est results in the quadratic equation
If b2 > 4 a c then there are two unique solutions for the quadratic equation and the general solution is
For the case of b2= 4 a c the general solution is
In the case of b2> 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.