The method of undetermined coefficients

In E12 we will extend what you learned in E11, but it will be more general. We will look at linear, time-invariant, continuous time, continuous variable, lumped parameter, dynamic physical systems. We will develop models for mechanical, thermal, biological... systems and show that they all yield differential equations that are of the form:

where y(t) is the output of the system, and f(t) is some known function.

**General Solutions method: **We solve for y(t) by assuming a solution of the form y(t)=y_{H}(t)+y_{P}(t).
The forms of the homogeneous and particular solution are given below.

Note: If f(t) is the sum of functions use superposition
(superposition states that if input f_{1}(t) yields output y_{1}(t), and input
f_{2}(t) yields output y_{2}(t) then the input f_{3}(t)=f_{1}(t)+f_{2}(t) will yield output
y_{3}(t)=y_{1}(t)+y_{2}(t)).

**Homogeneous Solution:** First solve for form of homogenous equation:

by assuming y_{H}(t)=Ae^{st} yields the
characteristic equation and homogeneous solution

The roots of characteristic equation determine allowable values of s, which
in turn determine the nature of the homogeneous response. Special
cases include repeated roots (If there are repeated
values of s, the homogenous solution also includes a time-multiplied
exponential. For example, if the characterstic
equation has repeated roots at s=-1, then y_{H}(t)=A_{1}e^{-t}+A_{2}te^{-t}) and complex
conjugate roots (damped sinusoids).

Initial conditions and the particular solution determine the values of A (we can't find these coefficients until after we know the particular response).

**Particular Solution:** Assume y_{P}(t) is of the same character as f(t)
(and all of its derivatives).

f(t) | y_{P}(t) |

A | K |

At | K_{1}t+K_{2} |

At^{2} |
K_{1}t^{2}+K_{2}t+K_{3} |

Ae^{at} |
Ke^{at} |

Acos(ω·t) | Csin(ω·t)+Dcos(ω·t) = Kcos(ω·t+φ) |

Asin(ω·t) | Csin(ω·t)+Dcos(ω·t) = Kcos(ω·t+φ) |

Ate^{at} |
K_{1}te^{at}+K_{2}e^{at} |

Note: if a term in y_{H}(t) also appears in y_{P}(t),
multiply the term in y_{P}(t) by "t". For example if
the differential equation has y_{H}(t)=Ae^{-2t} and f(t)=Be^{-2t}
then y_{P}(t)=Kte^{-2t}.

Note: we can't solve for

γ(t)=unit step function, constant for t≥0.

Particular solution:

Homogeneous Solution:

Example 2:

t=unit ramp.

Particular solution:

To solve for unknown coefficients equate like powers of t:

**Complete solution: **

The form of the homogeneous solution remains unchanged, so we can go right to finding the complete solution: