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Chapter 8: Q7P (page 422)

Solve the following differential equations by the methods discussed above and compare computer solutions.

(D2−5D+6)y=0

Short Answer

Expert verified

The solution is.y=c1e2x+c2e3x

Step by step solution

01

Given information 

A differential equation is given as.(D2−5D+6)y=0

02

Differential equation. 

A differential equation of the form (D−a)(D−b)y=0, a≠bhas general solution

y=c1eax+c2ebx.

03

Find the solution of given differential equation. 

Supposethat.D=ddx

Then

Dy=dydxDy=y'D2y=ddx(dydx)d2ydx2=y''

Given differential equation is.(D2−5D+6)y=0

Auxiliary equation is-

D2−5D+6=0(D−2)(D−3)=0D=2D=3

These are unequal roots.

⇒D=2,D=3

Differential equation becomes

⇒(D−2)(D−3)y=0

These are separable equation with solution

y=c1e2xandy=c2e3x

The general solution will be givenby.y=c1e2x+c2e3x

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Most popular questions from this chapter

In Problems 13 to 15, find a solution (or solutions) of the differential equation not obtainable by specializing the constant in your solution of the original problem. Hint: See Example 3.

13. Problem 2

(a) Show that D(eaxy)=eax(D+a)y,D2(eaxy)=eax(D+a)2y, and so on; that is, for any positive integral n, Dn(eaxy)=eax(D+a)ny.

Thus, show that ifis any polynomial in the operator D, then L(D)(eaxy)=eaxL(D+a)y.

This is called the exponential shift.

(b) Use to show that (D-1)3(exy)=exD3y,(D2+D-6)(e-3xy)=e-3x(D2-5D)y..

(c) Replace Dby D-a, to obtain eaxP(D)y=P(D-a)eaxy

This is called the inverse exponential shift.

(d) Using (c), we can change a differential equation whose right-hand side is an exponential times a

polynomial, to one whose right-hand side is just a polynomial. For example, consider

(D2-D-6)y=10×e2x; multiplying both sides by e-3xand using (c), we get

e-3x(D2-D-6)y=[D+32-D+3-6"]"ye-3x=(D2+5D)ye-3x=10x

Show that a solution of (D2+5D)u=10xis u=x2-25x; then or use this method to solve Problems 23 to 26.

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.yt-y=2tt,2t=3

Use L28 to find the Laplace transform of

f(t)={sin(t-π/2,t>π/20,t<π/2

Prove the general formula L29.

See all solutions

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