Kansas State University

Discussion

Paradigm

We obtain $D^2+4D+3=0$ which has roots $-1$ and $-3$. So the general solution to the homogeneous equation is $y_h(x)=c_1e^{-x}+c_2e^{-3x}$.

Step 2: Guess the form of the particular solution $y_p(x)$.

As indicated in the discussion above, a good guess for the form of the particular solution is $y_p(x)=Ae^{-2x}$.

Step 3: Plug the guess for the particular solution, $y_p(x)$, into the equation and solve for the undetermined coefficients.

We have $$\eqalign { y_p(x)&=Ae^{-2x} \cr y_p'(x)&=-2Ae^{-2x} \cr y_p''(x)&=4Ae^{-2x} \cr } $$ So plugging into the equation we find $$ \eqalign { y_p''(x)+4y_p'(x)+3y_p(x)&=4Ae^{-2x}+4(-2Ae^{-2x})+3(Ae^{-2x}) \cr &=(4A-8A+3A)e^{-2x} \cr &=-Ae^{-2x} \buildrel{\text{set}}\over{=} 4e^{-2x} \cr} $$ Solving the last equation we get $A=-4$ so $y_p(x)=-4e^{-2x}$.

Step 4:The general solution is now $y_p(x)+y_h(x)$.

The general solution to the equation is $$y(x)=-4e^{-2x}+c_1e^{-x}+c_2e^{-3x}$$ That is a very simple process. We now consider some of the situations where things can go wrong.

EXAMPLE 1: $$y''+4y'+3y=2e^{-x}$$ Step 1: $D^2+4D+3=0$ still has roots $-1$ and $-3$ and the general solution to the homogeneous equation is still $y_h(x)=c_1e^{-x}+c_2e^{-3x}$.

Step 2: Here the right hand side has the form $Be^{rx}$ where $r=-1$ is one of the roots of the homogeneous equation. Suppose we try our guess $y_p(x)=Ae^{-x}$ anyway. Then $y_p'(x)=-Ae^{-x}$ and $y_p''(x)=Ae^{-x}$ and so plugging into the equation yields $$ \eqalign { y_p''+4y_p'+3y_p&=Ae^{-x}+4(-Ae^{-x})+3(Ae^{-x}) \cr &=(A-4A+3A)e^{-x} \cr &=0 \buildrel{\text{set}}\over{=} 2e^{-x} \cr } $$ which is a problem. The trouble here is that since $e^{-x}$ is part of the homogeneous solution, it drops out. What we need is a second term of the form $e^{-x}$ which won't drop out. Looking back at section 6, we see that if we have a double root we have a term of the form $Axe^{rx}$ so let's try that in our present problem.

Step 2: (Take 2) Guess $y_p(x)=Axe^{-x}$

Step 3: $$ \eqalign { y_p(x)&=Axe^{-x} \cr y_p'(x)&=Ae^{-x}-Axe^{-x} \cr y_p''(x)&=-2Ae^{-x}+Axe^{-x} \cr } $$ So plugging into the equation we obtain $$ \eqalign { y_p''+4y_p'+3y_p&=(-2Ae^{-x}+Axe^{-x})+4(Ae^{-x}-Axe^{-x})+3(Axe^{-x}) \cr &=(-2A+4A)e^{-x}+(A-4A+3A)xe^{-x} \cr &=2Ae^{-x} \buildrel{\text{set}}\over{=} 2e^{-x} \cr} $$ Solving the last equation we find $A=1$ and $y_p(x)=xe^{-x}$.

Step 4: $y(x)=xe^{-x}+c_1e^{-x}+c_2e^{-3x}$.

There are two points to be noted about the last example. One is that if your first guess doesn't work, try try again. There is no penalty for making a wrong guess, beyond wasting a little bit of time, so try your guess and if it doesn't work, guess again. The second point is that you can often get some insight into the need for a second guess from what goes wrong with the first guess. In particular from this example, if you end up with an equation of the form $0=Be^{rx}$, you probably need to multiply your initial guess by $x$.

EXAMPLE 2: $$y''+4y'+4y=e^{-2x}$$ Step 1: We obtain $D^2+4D+4=(D+2)^2=0$ so we have a double root of $-2$ and the general solution of the homogeneous equation is $y_h(x)=c_1e^{-2x}+c_2xe^{-2x}$.

Step 2: We don't want to guess $Ae^{-2x}$ because that is part of the homogeneous solution. But so is $Axe^{-2x}$. So let's try $y_p(x)=Ax^2e^{-2x}$.

Step 3: $$ \eqalign { y_p(x)&=Ax^2e^{-2x} \cr y_p'(x)&=-2Ax^2e^{-2x}+2Axe^{-2x} \cr y_p''(x)&=4Ax^2e^{-2x}-8Axe^{-2x}+2Ae^{-2x} \cr } $$ So plugging into the equation yields $$ \eqalign { y_p''+4y_p'+4y_p&=(4Ax^2e^{-2x}-8Axe^{-2x}+2Ae^{-2x}) \cr &\qquad+4(-2Ax^2e^{-2x}+2Axe^{-2x})+4(Ax^2e^{-2x}) \cr &=(4A-8A+4A)x^2e^{-2x}+(-8A+8A)xe^{-2x}+2Ae^{-2x} \cr &=2Ae^{-2x} \buildrel{\text{set}}\over{=} e^{-2x} \cr } $$ Solving the last equation we find $A=1/2$ and so $y_p=1/2x^2e^{-2x}$.

Step 4: $y(x)=1/2x^2e^{-2x}+c_1e^{-2x}+c_2xe^{-2x}$.

Now we consider some right hand sides which are not quite of the form $Be^{rx}$, but not far from it.

EXAMPLE 3: $$y''+5y'+6y=2xe^x$$ Step 1: $D^2+5D+6=0$ has roots $-2$ and $-3$ so $y_h(x)=c_1e^{-2x}+c_2e^{-3x}$.

Step 2: Well, the first guess that probably comes to your mind is $y_p(x)=Axe^x$, so let's give that a try.

Step 3: $$ \eqalign { y_p(x)&=Axe^x \cr y_p'(x)&=Axe^x+Ae^x \cr y_p''(x)&=Axe^x+2Ae^x \cr } $$ Plugging into our equation we obtain $$ \eqalign { y_p''+5y_p'+6y_p&=(Axe^x+2Ae^x)+5(Axe^x+Ae^x)+6(Axe^x) \cr &=(A+5A+6)xe^x+(2A+5A)e^x \cr &=12Axe^x+7Ae^x \buildrel{\text{set}}\over{=} 2xe^x \cr } $$ Well that almost worked out. Unfortunately, we got that extra $7Ae^x$ term. So how should we get rid of it? But no real harm done, we just have to guess again. And we use the fact that we got an extra $7Ae^x$ term to guess that we need a $Be^x$ term to go along with our $Axe^x$ term. That makes sense. Whenever we've had an $xe^x$ term in the past in any solution we had an $e^x$ to go along with it.

Step 2: (Take 2) $y_p=Axe^x+Be^x$.

Step 3: (Take 2) $$ \eqalign { y_p(x)&=Axe^x+Be^x \cr y_p'(x)&=Axe^x+(A+B)e^x \cr y_p''(x)&=Axe^x+(2A+B)e^x \cr } $$ Plugging into our equation we obtain $$ \eqalign { y_p''+5y_p'+6y_p&=(Axe^x+(2A+B)e^x)+5(Axe^x+(A+B)e^x)+6(Axe^x+Be^x) \cr &=(A+5A+6A)xe^x+(2A+B+5A+5B+6B)e^x \cr &=12Axe^x+(7A+12B)e^x \buildrel{\text{set}}\over{=} 2xe^x \cr } $$ From the last equation we find $12A=2$ and $7A+12B=0$, so $A=1/6$ and $B=-7/72$. Hence $y_p(x)=1/6xe^x-7/72e^x$.

Step 4: $y(x)=1/6xe^x-7/72e^x+c_1e^{-2x}+c_2e^{-3x}$.

Before we consider the next example, we state and prove an easy and useful theorem (the best kind).

Theorem: Suppose $L$ is a linear operator and $Ly=f$ and $Lz=g$. Then $L(y+z)=f+g$.

Proof: This follows immediately from the definition of linearity. $L(y+z)=Ly+Lz=f+g$.

EXAMPLE 4: $$y''-2y'-8y=e^x-e^{-x}$$ Step 1: $D^2-2D-8=0$ has roots $-2$ and $4$ so $y_h(x)=c_1e^{-2x}+c_2e^{4x}$.

Step 2: Using the previous theorem, we can now split the hunt for a particular solution into two parts.

Step 2: (Part 1) Find a particular solution for $y''-2y'-8y=e^x$. We guess $y_{p1}(x)=Ae^x$.

Step 3: (Part 1) $$ \eqalign { y_{p1}(x)&=Ae^x \cr y_{p1}'(x)&=Ae^x \cr y_{p1}''(x)&=Ae^x \cr } $$ Plugging into our equation we obtain $$ \eqalign { y_{p1}''-2y_{p1}'-8y_{p1}&=Ae^x-2(Ae^x)-8(Ae^x) \cr &=(A-2A-8A)e^x \cr &=-9Ae^x \buildrel{\text{set}}\over{=} e^x \cr} $$ From the last equation we find $A=-1/9$ so $y_{p1}(x)=-1/9e^x$.

Step 2: (Part 2) Find a particular solution for $y''-2y'-8y=-e^{-x}$. We guess $y_{p2}(x)=Ae^{-x}$.

Step 3: (Part 2) $$ \eqalign { y_{p2}(x)&=Ae^{-x} \cr y_{p2}'(x)&=-Ae^{-x} \cr y_{p2}''(x)&=Ae^{-x} \cr } $$ Plugging into our equation we obtain $$ \eqalign { y_{p2}''-2y_{p2}'-8y_{p2}&=Ae^{-x}-2(-Ae^{-x})-8(Ae^{-x}) \cr &=(A+2A-8A)e^{-x} \cr &=-5Ae^{-x} \buildrel{\text{set}}\over{=} -e^{-x} \cr } $$ From the last equation we find $A=1/5$ so $y_{p2}(x)=1/5e^{-x}$.

Step 3: (Conclusion) $y_p(x)=y_{p1}(x)+y_{p2}(x)=-1/9e^x+1/5e^{-x}$.

Step 4: $y(x)=-1/9e^x+1/5e^{-x}+c_1e^{-2x}+c_2e^{4x}$.

Finally we consider how to handle sines and cosines on the right hand side. There are two different approaches and I will give examples of both. Most students find the real variable method easier than the complex method, but be warned that in later courses you may discover the complex method easier in many applications.

We will begin with the real variables approach. The key idea is to figure out what to guess if the right hand side is a sine and/or cosine. The key result is that whenever your right hand side is of the form $a\sin(\omega x)+b\cos(\omega x)$, guess a particular solution in the form $y_p(x)=A\sin(\omega x)+B\cos(\omega x)$. It needs to be emphasized that you must guess $y_p(x)$ with a sine/cosine pair, even if the original right hand side just has an individual sine or cosine. Just because $a$ or $b$ is 0 doesn't mean either $A$ or $B$ will be.

EXAMPLE 5: $$y''+3y'+2y=\sin(2x)$$ Step 1: We obtain $D^2+3D+2=0$ which has roots $-1$ and $-2$. So the general solution to the homogeneous equation is $y_h(x)=c_1e^{-x}+c_2e^{-2x}$.

Step 2: Since the right hand side is of the form $\sin(2x)$, we guess $y_p(x)=A\sin(2x)+B\cos(2x)$.

Step 3: $$ \eqalign { y_p(x)&=A\sin(2x)+B\cos(2x) \cr y_p'(x)&=2A\cos(2x)-2B\sin(2x) \cr y_p''(x)&=-4A\sin(2x)-4B\cos(2x) \cr }$$ Plugging into our equation we obtain $$ \eqalign { y_p''+3y_p'+2y_p&=(-4A-6B+2A)\sin(2x)+(-4B+6A+2B)\cos(2x) \cr &=(-2A-6B)\sin(2x)+(6A-2B)\cos(2x) \buildrel{\text{set}}\over{=} \sin(2x) \cr }$$ This gives us two equations in two unknowns, $$ \eqalign { -2A-6B&=1 \cr 6A-2B&=0 \cr } $$ Solving the pair of equations we get $A=-1/20$ and $B=-3/20$, so the particular solution is $y_p(x)=-1/20 \sin(2x)-3/20 \cos(2x)$.

Step 4: $y(x)=-1/20 \sin(2x)-3/20 \cos(2x)+c_1e^{-x}+c_2e^{-2x}$.

We will go over the complex method next. We start with a theorem similar to that in the previous section.

Theorem: If $a$, $b$ and $c$ are real and $ay''+by'+cy=f$, then $a(\Re[y])''+b(\Re[y])'+c(\Re[y])=\Re[f]$ and $a(\Im[y])''+b(\Im[y])'+c(\Im[y])=\Im[f]$.

Proof: Take the real and imaginary parts of each side of the given equation $ay''+by'+cy=f$.

This gives us a strategy to use with sines and cosines. We write them as the real part of a complex exponential and then find a particular solution for the complex equation. Finally we take the real part of the complex particular solution to find the particular solution to the original real equation. This differs enough from the first paradigm that I will write it down as a new paradigm.

Paradigm:

The roots of $D^2+D-2=0$ are $-2$ and $1$ so the general solution of the homogeneous equation is $y_h(x)=c_1e^{-2x}+c_2e^x$.

Step 2:Write the right hand side of the equation as the real part of a complex exponential.

$-\cos(2x)=\Re[-e^{i2x}]$.

Step 3: Find a particular solution to the corresponding complex equation.

Let $\tilde{y}_p$ denote a particular solution to $\tilde{y}_p''+\tilde{y}_p'-2\tilde{y}_p=-e^{i2x}$. Then we use the techniques developed earlier in this section to find $\tilde{y}_p(x)$.

SubStep 1: Guess the form of $\tilde{y}_p$

$\tilde{y}_p=Ae^{2ix}$.

SubStep 2: Plug into the equation and solve for the undetermined coefficients. $$ \eqalign { \tilde{y}_p&=Ae^{2ix} \cr \tilde{y}_p'&=2iAe^{2ix} \cr \tilde{y}_p''&=-4Ae^{2ix} \cr } $$ Plugging into the equation we obtain $$ \eqalign { \tilde{y}_p''+\tilde{y}_p'-2\tilde{y}_p&=(-4Ae^{2ix})+(2iAe^{2ix})-2(Ae^{2ix}) \cr &=(-4A+2iA-2A)e^{2ix} \cr &=(-6+2i)Ae^{2ix} \buildrel{\text{set}}\over{=} -e^{2ix} \cr } $$ Solving the last equation we find $A=-1/(-6+2i)=3/20+1/20i$. So $\tilde{y}_p(x)=(3/20+1/20i)e^{2ix}$.

Step 4: Take the real part of the complex particular solution to find the real particular solution.

$y_p(x)=\Re[\tilde{y}_p(x)]=3/20\cos(2x)-1/20\sin(2x)$.

Step 5: The general solution is $y(x)=y_p(x)+y_h(x)$.

$y(x)=3/20\cos(2x)-1/20\sin(2x)+c_1e^{-2x}+c_2e^x$.

We illustrate a few more possible difficulties in the final two examples.

EXAMPLE 6: $$y''+3y'+2y=3e^{-2x}\sin(x)$$ Step 1: The roots of $D^2+3D+2=0$ are $-1$ and $-2$ so the homogeneous solution is $y_h(x)=c_1e^{-x}+c_2e^{-2x}$.

Step 2: $3e^{-2x}\sin(x)=\Re[-3ie^{(-2+i)x}]$. (Note that we have to put the $-3i$ to get a $3$ after multiplying by $i\sin(x)$ in the identity for complex exponentials. It would also be possible to write the real function as the imaginary part of a complex exponential rather than the real part. In that case, we would have to take the imaginary part of the complex particular solution in step 4.)

Step 3:

SubStep 1: $\tilde{y}_p(x)=Ae^{(-2+i)x}$

SubStep 2: $$ \eqalign { \tilde{y}_p(x)&=Ae^{(-2+i)x} \cr \tilde{y}_p'(x)&=(-2+i)Ae^{(-2+i)x} \cr \tilde{y}_p''(x)&=(3-4i)Ae^{(-2+i)x} \cr } $$ Plugging into our equation we obtain $$ \eqalign { \tilde{y}_p''+3\tilde{y}_p'+2\tilde{y}_p&=[(3-4i)Ae^{(-2+i)x}] + 3[(-2+i)Ae^{(-2+i)x}]+2[Ae^{(-2+i)x}] \cr &=(3-4i-6+3i+2)Ae^{(-2+i)x} \cr &=(-1-i)Ae^{(-2+i)x} \buildrel{\text{set}}\over{=} -3ie^{(-2+i)x} \cr } $$ Solving the last equation we find $A=-3i/(-1-i)=3/2+3/2i$. So $\tilde{y}_p(x)=(3/2+3/2i)e^{(-2+i)x}$.

Step 4: $y_p(x)=e^{-2x}(3/2\cos(x)-3/2\sin(x))$.

Step 5: $y(x)=e^{-2x}(3/2\cos(x)-3/2\sin(x))+c_1e^{-x}+c_2e^{-2x}$.

EXAMPLE 7: $$y''+y=\cos(x)+2\sin(x)$$ Step 1: The roots of $D^2+1=0$ are $\pm i$ so the homogeneous solution is $y_h(x)=c_1\cos(x)+c_2\sin(x)$.

Step 2: We could split this into two pieces, but since both terms have the same frequency we can do it all in one by writing $\cos(x)+2\sin(x)=\Re[(1-2i)e^{ix}]$.

Step 3:

SubStep 1: $\tilde{y}_p(x)=Axe^{ix}$. Since $e^{ix}$ already appears in the homogeneous solution, we have to multiply by $x$.

SubStep 2: $$ \eqalign { \tilde{y}_p(x)&=Axe^{ix} \cr \tilde{y}_p'(x)&=iAxe^{ix}+Ae^{ix} \cr \tilde{y}_p''(x)&=-Axe^{ix}+2iAe^{ix} \cr } $$ Plugging into our equation we obtain $$ \eqalign { \tilde{y}_p''+\tilde{y}_p&=(-Axe^{ix}+2iAe^{ix})+(Axe^{ix}) \cr &=(-A+A)xe^{ix}+2iAe^{ix} \cr &=2iAe^{ix} \buildrel{\text{set}}\over{=} (1-2i)e^{ix} \cr } $$ Solving the last equation we find $A=(1-2i)/(2i)=-1-1/(2i)=-1+(1/2)i$. So $\tilde{y}_p(x)=(-1+(1/2)i)xe^{ix}$.

Step 4: $y_p(x)=-x\cos(x)+(1/2)x\sin(x)$.

Step 5: $y(x)=-x\cos(x)+1/2x\sin(x)+c_1\cos(x)+c_2\sin(x)$.

After all these examples, the reader may have discovered the following general rules.

• If the right-hand side is of the form $e^{rx}$ and $e^{rx}$ is not already in the homogeneous solution, then the guess should be $Ae^{rx}$.

• If the right-hand side is of the form $(a_nx^n+\cdots a_1x+a_0)e^{rx}$ with $a_n\ne0$ and $e^{rx}$ is not already in the homogeneous solution, then the guess should be $(A_nx^n+\cdots A_1x+A_0)e^{rx}.$ Note that while $a_n\ne0$, the other $a_j$ terms may or may not be 0 in this case.

• If the right-hand side is of the form $(a_nx^n+\cdots a_1x+a_0)e^{rx}$ with $a_n\ne0$ and $x^me^{rx}$ is in the homogeneous solution, then the guess should be $x^{m+1}(A_nx^n+\cdots A_1x+A_0)e^{rx}.$

• If the right-hand side is of the form $a\sin(\omega x)+b\cos(\omega x)$, guess $A\sin(\omega x)+B\cos(\omega x)$. Note that you have to include both $A$ and $B$ even if $a$ or $b$ is 0.

• You can also replace sines and cosines with complex exponentials to apply the above rules to them.

You can generate additional examples of second-order constant-coefficient linear inhomogeneous initial value problems here.