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Chapter 8: Q8-2-15P (page 398)

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

Problem 11

Short Answer

Expert verified

The solution is y=2.

Step by step solution

01

Given Information.

Differential Equation given is 2y'=3y-213

02

Definition of Differential equation

A differential equation is an equation that contains at least onederivativeof an unknown function, either an ordinary derivative or a partial derivative.

03

Solve differential equation

Separate the variables in problem 11 to get

dyy-213=32dx

The general solution of this differential equation is

y=x+C132+2

Now,dyy2is not valid for y=2

y=2 is a solution of this differential equation that cannot be obtained by any choice of C1

Therefore, the solution is y=2 .

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Most popular questions from this chapter

Using (3.9), find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after (3.9), and Example .

y+y/x2+1=1/(x+x2+1)

(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.

Heat is escaping at a constant rate [dQdtin (1.1)is constant] through the walls of a long cylindrical pipe. Find the temperature T at a distance r from the axis of the cylinder if the inside wall has radius r=1and temperature T=100and the outside wall has r=2and T=0

Find the orthogonal trajectories of each of the following families of curves. In each case, sketch or computer plot several of the given curves and several of their orthogonal trajectories. Be careful to eliminate the constant from y'for the original curves; this constant takes different values for different curves of the original family, and you want an expression for y'which is valid for all curves of the family crossed by the orthogonal trajectory you are trying to find. See equations (2.10)to (2.12)

x2+y2=cost.

Solve (12.3)if G=0and dG/dt=0at t=0 to obtain (12.5). Hint: Use L28 and L3 to find the inverse transform.
See all solutions

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