Details

Snapshot 1: the case , , , corresponds to simple harmonic motion

Snapshot 2: the case where is a quadratic function and

Snapshot 3: a failure case: , hence the complementary function is of the form , and since , the particular integral is of the form , where is a constant

References

[1] L. Bostock, S. Chandler, and C. Rourke, Further Pure Mathematics, Cheltenham, UK: Stanley Thornes, 1990 pp. 274–285.

[2] J. Berry, T. Graham, R. Porkess, and P. Mitchell, MEI Structured Mathematics: Differential Equations, 3rd ed., Abingdon, UK: Hodder Murray, 2006 pp. 74–141.