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Mathematics LibreTexts

13.5: Differential equations

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Coverup Method

We are going to use partial fractions and the coverup method. We will assume you have seen partial fractions. If you don’t remember them well or have never seen the coverup method.

Example 13.5.1

Solve yy=e2t, y(0)=1, y(0)=1 using Laplace transform.

Solution

Call L(y)=Y. Apply the Laplace transform to the equation gives

(s2Ysy(0)y(0))Y=1s2

A little bit of algebra now gives

(s21)Y=1s2+s+1.

So

Y=1(s2)(s21)+s+1s21=1(s2)(s21)+1s1

Use partial fractions to write

Y=As2+Bs1+Cs+1+1s1.

The coverup method gives A=1/3,B=1/2,C=1/6.

We recognize

1sa

as the Laplace transform of eat, so

y(t)=Ae2t+Bet+Cet+et=13e2t12et+16et+et.

Example 13.5.2

Solve yy=1, y(0)=0, y(0)=0.

Solution

The rest (zero) initial conditions are nice because they will not add any terms to the algebra. As in the previous example we apply the Laplace transform to the entire equation.

s2YY=1s, so Y=1s(s21)=1s(s1)(s+1)=As+Bs1+Cs+1

The coverup method gives A=1,B=1/2,C=1/2. So,

y=A+Bet+Cet=1+12et+12et.


This page titled 13.5: Differential equations is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.

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