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Chapter 8: Q17P (page 436)

Solve the following equations using method (d) above. x2y''+xy'-16y=8x4

Short Answer

Expert verified

The general solution of the equation isy=c1x-4+c2x4+(ln鈦�x)x4

Step by step solution

01

Given information

The given differential equation is x2y''+3xy'-3y=8x4.

02

Auxiliary equation

Auxiliary equation is an algebraic equation of degree nupon which depends the solution of a given nth-order differential equation or difference equation.

03

Solve for double differential

Consider the given equation.

x2y''+3xy'-3y=0Letx=ez.So,x=ezz=ln鈦�x

dzdx=1x

Now,

And,

xdydx=dydz

04

Roots of equation

The given differential equation can be written as,

(D(D-1)+D-16)y=8e4z

The auxiliary equation of the above equation is,

m(m-1)+m-16=0

The solution of the auxiliary equation is,

m=卤4

Let the roots be represented as,

a=-4b=4

05

Solve for Yp

Now,

Q=8e4z

So,

Now,

Drepresent the first derivative and D2represent the second derivative.

So,

06

Complete solution

Thus, the complete solution is given as,

Therefore, the general solution of the equation is y=c1x-4+c2x4+(ln鈦�x)x4

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Most popular questions from this chapter

Find the distance which an object moves in time tif it starts from rest and has accelerationd2xdt2=ge鈭�kt. Show that for smalltthe result is approximately, and for very large, the speeddxdtis approximately (1.10)constant. The constant is called the terminal speed . (This problem corresponds roughly to the motion of a parachutist.)

If P dollars are left in the bank at interest I percent per year compounded continuously, find the amount A at time t. Hint: Find dA, the interest on A dollars for time dt.

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 1.

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a) Show that鈭�鈰�(er/r2)=0forr>0.

(b) Show that鈭�(1/r)=-er/r2.
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