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Chapter 2: Q18E (page 64)

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

If it is, then solve it.

2x+y2-cos(x+y)dx+2xy-cos(x+y)-eydy=0

Short Answer

Expert verified

The solution is x2-sin(x+y)+xy2-ey=C.

Step by step solution

01

Evaluate whether the equation is exact                                                                                                                                                                                

Here 2x+y2-cos(x+y)dx+2xy-cos(x+y)-eydy=0

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

M(x,y)=2x+y2-cos(x+y)N(x,y)=2xy-cos(x+y)-ey

鈭�M鈭�y=2y+sin(x+y)=鈭�N鈭�x

This equation is exact.

02

Find the value of F(x, y)

Here

M(x,y)=2x+y2-cos(x+y)F(x,y)=鈭�M(x,y)dx+g(y)=鈭�2x+y2-cos(x+y)dx+g(y)=x2-sin(x+y)+g(y)

03

Determine the value of g(y)

鈭�F鈭�y(x,y)=N(x,y)-cos(x+y)+g'(y)=2xy-cos(x+y)-eyg'(y)=2xy-eyg(y)=xy2-ey+C1

NowF(x,y)=x2-sin(x+y)+xy2-ey+C

The solution of the differential equation is x2-sin(x+y)+xy2-ey=C.

Hence the solution is x2-sin(x+y)+xy2-ey=C

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Most popular questions from this chapter

Use the method discussed under 鈥淗omogeneous Equations鈥� to solve problems 9-16.

鈥�dy诲胃=胃sec(y胃)+y胃

Question: Use the substitution v=x-y+2 to solve equation (8).

dydx=y-x-1+x-y+2-1

In problems, 1-8 identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the formy'=Gax+by.t+x+2dx+3t-x-6dt=0

Use the method discussed under 鈥淗omogeneous Equations鈥� to solve problems 9-16鈥�xy+y2鈥�dx-x2dy=0

Question: Riccati Equation. An equation of the form (18) dydx=P(x)y2+Q(x)y+R(x) is called a generalized Riccati equation.

  1. If one solution鈥攕ay, u(x)鈥攐f (18) is known, show that the substitutiony=u+1vreduces (18) to a linear equation in v.
  2. Given thatux=xis a solution todydx=x3(y-x)2+yx,

use the result of part (a) to find all the other solutions to this equation. (The particular solutionux=xcan be found by inspection or by using a Taylor series method; see Section 8.1.)
See all solutions

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