Now we consider second order equations of the form $\ds a\ddot y+b\dot y+cy=f(t)$, with $a$, $b$, and $c$ constant. Of course, if $a=0$ this is really a first order equation, so we assume $a\not=0$. Also, much as in exercise 20 of section 19.5, if $c=0$ we can solve the related first order equation $\ds a\dot h+bh=f(t)$, and then solve $\ds h=\dot y$ for $y$. So we will only examine examples in which $c\not=0$.

Suppose that $\ds y_1(t)$ and $\ds y_2(t)$ are solutions to $\ds a\ddot y+b\dot y+cy=f(t)$, and consider the function $\ds h=y_1-y_2$. We substitute this function into the left hand side of the differential equation and simplify: $$ a(y_1-y_2)''+b(y_1-y_2)'+c(y_1-y_2)=ay_1''+by_1'+cy_1 - (ay_2''+by_2'+cy_2)=f(t)-f(t)=0. $$ So $h$ is a solution to the homogeneous equation $\ds a\ddot y+b\dot y+cy=0$. Since we know how to find all such $h$, then with just one particular solution $\ds y_2$ we can express all possible solutions $\ds y_1$, namely, $\ds y_1=h+y_2$, where now $h$ is the general solution to the homogeneous equation. Of course, this is exactly how we approached the first order linear equation.

To make use of this observation we need a method to find a single
solution $y_2$. This turns out to be somewhat more difficult than the
first order case, but if $f(t)$ is of a certain simple form, we can
find a solution using the
**method of undetermined coefficients**,
sometimes
more whimsically called the
**method of judicious guessing**.

Example 19.6.1 Solve the differential equation $\ds \ddot y-\dot y-6y=18t^2+5$. The general solution of the homogeneous equation is $\ds Ae^{3t}+Be^{-2t}$. We guess that a solution to the non-homogeneous equation might look like $f(t)$ itself, namely, a quadratic $\ds y=at^2+bt+c$. Substituting this guess into the differential equation we get $$ \ddot y-\dot y-6y = 2a-(2at+b)-6(at^2+bt+c) = -6at^2+(-2a-6b)t+(2a-b-6c). $$ We want this to equal $18t^2+5$, so we need $$\eqalign{ -6a&=18\cr -2a-6b&=0\cr 2a-b-6c&=5\cr} $$ This is a system of three equations in three unknowns and is not hard to solve: $a=-3$, $b=1$, $c=-2$. Thus the general solution to the differential equation is $\ds Ae^{3t}+Be^{-2t}-3t^2+t-2$.

So the "judicious guess'' is a function with the same form as $f(t)$ but with undetermined (or better, yet to be determined) coefficients. This works whenever $f(t)$ is a polynomial.

Example 19.6.2 Consider the initial value problem $\ds m\ddot y +ky=-mg$, $y(0)=2$, $\ds\dot y(0)=50$. The left hand side represents a mass-spring system with no damping, i.e., $b=0$. Unlike the homogeneous case, we now consider the force due to gravity, $-mg$, assuming the spring is vertical at the surface of the earth, so that $g=980$. To be specific, let us take $m=1$ and $k=100$. The general solution to the homogeneous equation is $\ds A\cos(10t)+B\sin(10t)$. For the solution to the non-homogeneous equation we guess simply a constant $y=a$, since $-mg=-980$ is a constant. Then $\ds \ddot y+100y= 100a$ so $a=-980/100=-9.8$. The desired general solution is then $\ds A\cos(10t)+B\sin(10t)-9.8$. Substituting the initial conditions we get $$\eqalign{ 2&=A-9.8\cr 50&=10B\cr} $$ so $A=11.8$ and $B=5$ and the solution is $\ds 11.8\cos(10t)+5\sin(10t)-9.8$.

More generally, this method can be used when a function similar to $f(t)$ has derivatives that are also similar to $f(t)$; in the examples so far, since $f(t)$ was a polynomial, so were its derivatives. The method will work if $f(t)$ has the form $p(t)e^{\alpha t}\cos(\beta t)+ q(t)e^{\alpha t}\sin(\beta t)$, where $p(t)$ and $q(t)$ are polynomials; when $\alpha=\beta=0$ this is simply $p(t)$, a polynomial. In the most general form it is not simple to describe the appropriate judicious guess; we content ourselves with some examples to illustrate the process.

Example 19.6.3 Find the general solution to $\ds\ddot y+7\dot y+10y=e^{3t}$. The characteristic equation is $r^2+7r+10=(r+5)(r+2)$, so the solution to the homogeneous equation is $Ae^{-5t}+Be^{-2t}$. For a particular solution to the inhomogeneous equation we guess $Ce^{3t}$. Substituting we get $$ 9Ce^{3t}+21Ce^{3t}+10Ce^{3t}=e^{3t}40C. $$ When $C=1/40$ this is equal to $f(t)=e^{3t}$, so the solution is $Ae^{-5t}+Be^{-2t}+(1/40)e^{3t}$.

Example 19.6.4 Find the general solution to $\ds\ddot y+7\dot y+10y=e^{-2t}$. Following the last example we might guess $Ce^{-2t}$, but since this is a solution to the homogeneous equation it cannot work. Instead we guess $Cte^{-2t}$. Then $$ (-2Ce^{-2t}-2Ce^{-2t}+4Cte^{-2t})+7(Ce^{-2t}-2Cte^{-2t})+10Cte^{-2t} =e^{-2t}(-3C). $$ Then $C=-1/3$ and the solution is $Ae^{-5t}+Be^{-2t}-(1/3)te^{-2t}$.

In general, if $f(t)=e^{kt}$ and $k$ is one of the roots of the characteristic equation, then we guess $Cte^{kt}$ instead of $Ce^{kt}$. If $k$ is the only root of the characteristic equation, then $Cte^{kt}$ will not work, and we must guess $Ct^2e^{kt}$.

Example 19.6.5 Find the general solution to $\ds\ddot y-6\dot y+9y=e^{3t}$. The characteristic equation is $\ds r^2-6r+9=(r-3)^2$, so the general solution to the homogeneous equation is $Ae^{3t}+Bte^{3t}$. Guessing $Ct^2e^{3t}$ for the particular solution, we get $$ (9Ct^2e^{3t}+6Cte^{3t}+6Cte^{3t}+2Ce^{3t})-6(3Ct^2e^{3t}+2Cte^{3t})+9Ct^2e^{3t} =e^{3t}2C. $$ The solution is thus $\ds Ae^{3t}+Bte^{3t}+(1/2)t^2e^{3t}$.

It is common in various physical systems to encounter an $f(t)$ of the form $\ds a\cos(\omega t)+b\sin(\omega t)$.

Example 19.6.6 Find the general solution to $\ds\ddot y+6\dot y+25y=\cos(4t)$. The roots of the characteristic equation are $-3\pm 4i$, so the solution to the homogeneous equation is $\ds e^{-3t}(A\cos(4t)+B\sin(4t))$. For a particular solution, we guess $C\cos(4t)+D\sin(4t)$. Substituting as usual: $$\eqalign{ (-16C\cos(4t)&+-16D\sin(4t))+6(-4C\sin(4t)+4D\cos(4t))+25(C\cos(4t)+D\sin(4t))\cr &=(24D+9C)\cos(4t)+(-24C+9D)\sin(4t).\cr} $$ To make this equal to $\cos(4t)$ we need $$\eqalign{ 24D+9C&=1\cr 9D-24C&=0\cr} $$ which gives $C=1/73$ and $D=8/219$. The full solution is then $\ds e^{-3t}(A\cos(4t)+B\sin(4t))+(1/73)\cos(4t)+(8/219)\sin(4t)$.

The function $\ds e^{-3t}(A\cos(4t)+B\sin(4t))$ is a damped
oscillation as in example 19.5.3,
while $\ds(1/73)\cos(4t)+(8/219)\sin(4t)$ is a simple undamped
oscillation. As $t$ increases, the sum $\ds
e^{-3t}(A\cos(4t)+B\sin(4t))$ approaches zero, so the solution
$$e^{-3t}(A\cos(4t)+B\sin(4t))+(1/73)\cos(4t)+(8/219)\sin(4t)$$
becomes more and more like the simple oscillation
$\ds(1/73)\cos(4t)+(8/219)\sin(4t)$—notice that the initial
conditions don't matter to this long term behavior. The damped portion
is called the
**transient** part of the
solution, and the simple oscillation is called the **steady state**
part of the solution.
A physical example is a mass-spring system. If the only force on the
mass is due to the spring, then the behavior of the system is a damped
oscillation. If in addition an external force is applied to the mass,
and if the force varies according to a function of the form
$\ds a\cos(\omega t)+b\sin(\omega t)$, then the long term behavior
will be a simple oscillation determined by the steady state part of the
general solution; the initial position of the mass will not matter.

As with the exponential form, such a simple guess may not work.

Example 19.6.7 Find the general solution to $\ds\ddot y+16y=-\sin(4t)$. The roots of the characteristic equation are $\pm4i$, so the solution to the homogeneous equation is $A\cos(4t)+B\sin(4t)$. Since both $\cos(4t)$ and $\sin(4t)$ are solutions to the homogeneous equation, $C\cos(4t)+D\sin(4t)$ is also, so it cannot be a solution to the non-homogeneous equation. Instead, we guess $Ct\cos(4t)+Dt\sin(4t)$. Then substituting: $$\eqalign{ (-16Ct\cos(4t)&-16D\sin(4t)+8D\cos(4t)-8C\sin(4t)))+16(Ct\cos(4t)+Dt\sin(4t))\cr &=8D\cos(4t)-8C\sin(4t).\cr} $$ Thus $C=1/8$, $D=0$, and the solution is $\ds C\cos(4t)+D\sin(4t)+(1/8)t\cos(4t)$.

In general, if $f(t)=a\cos(\omega t)+b\sin(\omega t)$, and $\pm \omega i$ are the roots of the characteristic equation, then instead of $C\cos(\omega t)+D\sin(\omega t)$ we guess $Ct\cos(\omega t)+Dt\sin(\omega t)$.

## Exercises 19.6

Find the general solution to the differential equation.

**Ex 19.6.1**
$\ds\ddot y -10\dot y+25y=\cos t$
(answer)

**Ex 19.6.2**
$\ds\ddot y+2\sqrt2\dot y+2y=10$
(answer)

**Ex 19.6.3**
$\ds\ddot y+16y=8t^2+3t-4$
(answer)

**Ex 19.6.4**
$\ds\ddot y+2y=\cos(5t)+\sin(5t)$
(answer)

**Ex 19.6.5**
$\ds\ddot y-2\dot y+2y=e^{2t}$
(answer)

**Ex 19.6.6**
$\ds\ddot y-6y+13=1+2t+e^{-t}$
(answer)

**Ex 19.6.7**
$\ds\ddot y+\dot y-6y=e^{-3t}$
(answer)

**Ex 19.6.8**
$\ds\ddot y-4\dot y+3y=e^{3t}$
(answer)

**Ex 19.6.9**
$\ds\ddot y+16y=\cos(4t)$
(answer)

**Ex 19.6.10**
$\ds\ddot y +9y=3\sin(3t)$
(answer)

**Ex 19.6.11**
$\ds\ddot y+12\dot y+36y=6e^{-6t}$
(answer)

**Ex 19.6.12**
$\ds\ddot y-8\dot y+16y=-2e^{4t}$
(answer)

**Ex 19.6.13**
$\ds\ddot y+6\dot y+5y=4$
(answer)

**Ex 19.6.14**
$\ds\ddot y-\dot y-12y=t$
(answer)

**Ex 19.6.15**
$\ds\ddot y+5y=8\sin(2t)$
(answer)

**Ex 19.6.16**
$\ds\ddot y-4y=4e^{2t}$
(answer)

Solve the initial value problem.

**Ex 19.6.17**
$\ds\ddot y-y=3t+5$, $y(0)=0$, $\ds\dot y(0)=0$
(answer)

**Ex 19.6.18**
$\ds\ddot y+9y=4t$, $y(0)=0$, $\ds\dot y(0)=0$
(answer)

**Ex 19.6.19**
$\ds\ddot y +12\dot y +37y=10e^{-4t}$, $y(0)=4$, $\ds\dot y(0)=0$
(answer)

**Ex 19.6.20**
$\ds\ddot y+6\dot y+18y=\cos t-\sin t$, $y(0)=0$, $\ds\dot
y(0)=2$
(answer)

**Ex 19.6.21**
Find the solution for the mass-spring equation
$\ds\ddot y+4\dot y+29y=689\cos(2t)$.
(answer)

**Ex 19.6.22**
Find the solution for the mass-spring equation
$\ds3\ddot y+12\dot y+24y=2\sin t$.
(answer)

**Ex 19.6.23**
Consider the differential
equation $\ds m\ddot y+b\dot y+ky=\cos(\omega t)$,
with $m$, $b$, and $k$ all positive and $\ds b^2< 2mk$; this equation
is a model for a damped mass-spring system with external
driving force $\cos(\omega t)$.
Show that the steady state part of the solution has amplitude
$${1\over \sqrt{(k-m\omega^2)^2+\omega^2b^2}}.$$
Show that this amplitude is largest when
$\ds \omega={\sqrt{4mk-2b^2}\over 2m}$. This is the
**resonant frequency** of the system.