Solving Higher Order Differential Equations
I. Homogeneous Linear Equations (those = 0)
1. Use the coefficients to set up and auxiliary equation equal to zero and
solve using factoring, the quadratic formula or synthetic division.
2. Use those solutions to set up the general solution.
a. If they are real and different, then y = c1e--- + c2e --b. If they are real and alike, then y = c1e--- + c2te--c. If they are imaginary, then y = e----(c1 cos (_t) + c2 sin (__t)
II. Undetermined coefficients
1. Find the complementary solution in the method listed in I.
2. Use the right hand side to set up sets. Eliminate those that are
alike, and multiply any of the sets that match the solutions of yc
by a power of the independent variable.
3. Write what you have left as a sum (linear combination) with
coefficients of A,B,C
4. Find the first and second derivatives of this sum and substitute in
the original equations
5. Set equal expressions together and solve for the A,B,C
6. Write a linear combination of the complementary function and the
particular integral.
III. Variation of parameters.
1. Do the left side as if homogeneous to obtain yc.
2. Rewrite the solution as y = v1(x) y1(x) + v2(x)y2(x)
3. Find the first derivative and set the v’(x)’s equal to zero.
4. Take the second derivative.
5. Substitute into the original equation and get 2nd equation in terms
of v’(x).
6. Use Cramer’s Rule and integration to find v1(x) and v2(x).
7. Substitute these expressions back into step 2.
8. Write the general solution as a linear combination of yc and yp.
IV. Cauchy-Euler
1. Let x = et or ln x = t ; dy = 1 dy ; d2y = 1 ( d2y - dy)
dx x dt dx2 x2 dt2 dt
2. Substitute in the original equation.
3. Find the solution by Homogeneous, UC or Variation of parameters.
4. Change your solution back in terms of x.
V. Reduction of order.
1. Let y = v f(x) where f(x) is your given solution.
2. Find the first and second derivatives and substitute them into the
original equation.
3. Let w = dv\dx and dw\dx = d2v\dx2 and solve the first order d.e.
4. Substitute in dv\dx for w and integrate to find v.
5. Write the general solution by multiplying the value of v time the
given solution.