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Chapter 1: Q25E (page 1)

In Problems 21鈥�26, solve the initial value problem.

(y2sinx)dx+(1x-yx)dy,y(蟺)=1

Short Answer

Expert verified

The solution is sinx-xcosx=lny+1y+蟺-1.

Step by step solution

01

     Step 1: Evaluate the equation is exact               

Here (y2sinx)dx+1x-yxdy,y(蟺)=1

The condition for exact is 鈭�M鈭�y=鈭�N鈭�x.

M(x,y)=(y2sinx)N(x,y)=1x-yx鈭�M鈭�y=2ysinx鈭�N鈭�x=y-1x2鈭�M鈭�y鈮�鈭�N鈭�x

This equation is not exact.

02

Find the solution

But this equation can be separable.

(y2sinx)dx+1-yxdy=0(xsinx)dx=y-1y2dy

Integrating on both sides, the result is

-xcosx+sinx=lny+1y+C

Therefore, the solution of the differential equation is-xcosx+sinx=lny+1y+C

03

Apply the initial conditions

Apply the initial conditionsy(蟺)=1.

-蟺cos蟺+sin蟺=ln1+11+CC=蟺-1

The solutions is

.

-xcosx+sinx=lny+1y+蟺-1sinx-xcosx=lny+1y+蟺-1

Hence the solution is role="math" localid="1664190043647" sinx-xcosx=lny+1y+蟺-1

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Most popular questions from this chapter

In Problems 9-13, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship implicitly defines y as a function of x and use implicit differentiation.

siny+xy-x3=2,y''=6xy'+y'3siny-2y'23x2-y

Question: The Taylor series for f(x) =ln (x)about x2=0given in equation (13) can also be obtained as follows:

(a)Starting with the expansion 1/ (1-s) =鈭�n=0鈭�s'' and observing that

'

obtain the Taylor series for 1/xabout x0= 1.

(b)Since use the result of part (a) and termwise integration to obtain the Taylor series for f (x)=lnxaboutx0= 1.

In Problems 13-19,find at least the first four nonzero terms in a power series expansion of the solution to the given initial value problem.

(x2+1)y''-exy'+y=0;y(0)=1,y'(0)=1

The logistic equation for the population (in thousands) of a certain species is given by dpdt=3p-2p2.

猞� Sketch the direction field by using either a computer software package or the method of isoclines.

猞� If the initial population is 3000 [that is, p(0) = 3], what can you say about the limiting population?

猞� If p(0)=0.8, what is limt鈫�+鈭�p(t)?

猞� Can a population of 2000 ever decline to 800?

In Problems 21鈥�26, solve the initial value problem.

(ety+tety)dt+(tet+2)dy=0,y(0)=-1
See all solutions

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