Differential Equations Higher-Order D.E. with constant coefficients Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
The general solution will always have the form: Here is the general form of an n-th order linear equation with constant coefficients. The general solution will always have the form: yh is the solution to the corresponding homogeneous equation, where g(t)=0 yp is a particular solution to the original DE. To find the homogeneous solution we can find the roots of the characteristic equation, just like we did for 2nd-order equations. The particular solution can be found by undetermined coefficients if the g(t) function has a convenient form, or a generalized version of variation of parameters. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Here are a few examples: The solution depends on being able to find the roots of the characteristic equation. If it factors nicely this is no problem, but can be tricky for higher orders. Here are a few examples: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Example 1 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
The characteristic equation for this one is: Example 1 The characteristic equation for this one is: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
The characteristic equation for this one is: Example 1 The characteristic equation for this one is: The trick is to recognize that this only has even powers, so we can use the quadratic formula, thinking of r2 as our variable. This one factors out nicely: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
The characteristic equation for this one is: Example 1 The characteristic equation for this one is: The trick is to recognize that this only has even powers, so we can use the quadratic formula, thinking of r2 as our variable. This one factors out nicely: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
The characteristic equation for this one is: Example 1 The characteristic equation for this one is: The trick is to recognize that this only has even powers, so we can use the quadratic formula, thinking of r2 as our variable. This one factors out nicely: There are 4 real roots, so the solution is: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Here is the characteristic equation: Example 2 Here is the characteristic equation: r=0 is definitely a solution, as is r=2. But there should be 4 solutions! How do you get the other two? They are actually complex numbers! We have a couple of options from here. Option 1: Long division Option 2: Convert to Polar form Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Since r=2 solves the equation, (r-2) must be a factor of (r3-8). Example 2 Option 1: Long Division Since r=2 solves the equation, (r-2) must be a factor of (r3-8). Divide out the factor (r-2) to find the remaining quadratic expression. Now use the quadratic formula on the remaining expression. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Option 2: Convert to Polar form Example 2 Option 2: Convert to Polar form Any complex number can be written in polar form with these formulas: In our problem we want to write the number 8 in polar form. imaginary 2∏/3 real 4∏/3 The equation we want to solve is: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Here is the characteristic equation: Example 2 Here is the characteristic equation: There are 4 roots: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Example 3 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
First solve the homogeneous equation: Example 3 First solve the homogeneous equation: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
First solve the homogeneous equation: Example 3 First solve the homogeneous equation: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
First solve the homogeneous equation: Example 3 First solve the homogeneous equation: Next make a guess for the particular solution: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
First solve the homogeneous equation: Example 3 First solve the homogeneous equation: Next make a guess for the particular solution: Comparing to yh notice that the Aet term is a solution to the homogeneous problem. So we have to modify our guess. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
The homogeneous solution is Example 3 The homogeneous solution is The form for the particular solution is Find the derivatives of yp then substitute to find the constants A, B and C Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB