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Chapter 1: Q13E (page 1)

In Problems 9鈥�20, determine whether the equation is exact.

If it is, then solve it.

et(y - t)dt +(1 +et)dy = 0

Short Answer

Expert verified

The solution isy=(t-1)et+C1+et.

Step by step solution

01

Evaluate whether the equation is exact

Hereety - tdt +1 +etdy = 0

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

M(t,y)=et(y-t)N(t,y)=(1+et)

鈭�M鈭�y=et=鈭�N鈭�t

This equation is exact.

02

Find the value of F (x, t)

Here

M(t,y)=et(y-t)F(t,y)=鈭�M(t,y)dt+g(y)=鈭�et(y-t)dt+g(y)=ety-ett+et+g(y)

03

Determine the value of g(y)

鈭�F鈭�y(t,y)=N(t,y)et+g'(y)=1+etg'(y)=1g(y)=y+C1

NowF(t,y) =ety -ett +et+ y +C1

The general solution of the differential equation isy=(t-1)et+C1+et

Hence the solution isrole="math" localid="1664172391787" y=(t-1)et+C1+et

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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.

x2+y2=4,dydx=xy

In problems 1-6, identify the independent variable, dependent variable, and determine whether the equation is linear or nonlinear.

y3d2xdy2+3x-8y-1=0

Using the Runge鈥揔utta algorithm for systems with h = 0.05, approximate the solution to the initial value problem y'''+y''+y2=t;y(0)=1,y'(0)=0,y''(0)=1 at t=1.

Verify thatx2+cy2=1, where c is an arbitrary non-zero constant, is a one-parameter family of implicit solutions todydx=xyx2-1 and graph several of the solution curves using the same coordinate axes.

Use the method in Problem 32 to find the orthogonal trajectories for each of the given families of curves, where k is a parameter.

(a)2x2+y2=k

(b)y=kx4

(c)y=ekx

(d)y2=kx

[Hint: First express the family in the form F(x, y) = k.]
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