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Chapter 1: Q9E (page 1)

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

If it is, then solve it.

(2xy+3)dx+(x2-1)dy=0

Short Answer

Expert verified

The solution isy=C-3x/x2-1.

Step by step solution

01

Evaluate whether the equation is exact

Here(2xy+3)dx+(x2-1)dy=0

The condition for exact is∂M∂y=∂N∂x .

M(x,y)=2xy+3N(x,y)=x2-1∂M∂y=2x=2x=∂N∂x

This equation is exact.

02

Find the value of F(x, y)

Here

M(x,y)=2xy+3F(x,y)=∫M(x,y)dx+g(y)=∫(2xy+3)dx+g(y)=x2y+3x+g(y)

03

Determine the value of g(y)

∂F∂y(x,y)=N(x,y)x2+g'(y)=x2-1g'(y)=-1g(y)=-y

NowF(x,y)=x2y+3x-y

x2y+3x-y=C(x2-1)y+3x=C(x2-1)y=C-3xy=(C-3x)/(x2-1)

Therefore, the solution isy=(C-3x)/(x2-1)y=(C-3x)/(x2-1)

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Most popular questions from this chapter

Consider the differential equation dydx=x+siny

⦁ A solution curve passes through the point (1,π2). What is its slope at this point?

⦁ Argue that every solution curve is increasing for x>1.

⦁ Show that the second derivative of every solution satisfies d2ydx2=1+xcosy+12sin2y.

⦁ A solution curve passes through (0,0). Prove that this curve has a relative minimum at (0,0).

Find a general solution for the differential equation with x as the independent variable:

y'''+3y''−4y'−6y=0

Implicit Function Theorem. Let G(x,y)have continuous first partial derivatives in the rectangleR={x,y:a<x<b,c<y<d}containing the pointlocalid="1664009358887" (x0,y0). IfG(x0,y0)=0 and the partial derivativeGy(x0,y0)≠0, then there exists a differentiable function y=ϕ(x), defined in some intervalI=(x0-δ,y0+δ),that satisfies G for allforG(x,ϕx)all x∈I.

The implicit function theorem gives conditions under which the relationshipG(x,y)=0 implicitly defines yas a function of x. Use the implicit function theorem to show that the relationshipx+y+exy=0 given in Example 4, defines y implicitly as a function of x near the point(0,-1).

Consider the question of Example 5 ydydx-4x=0

  1. Does Theorem 1 imply the existence of a unique solution to (13) that satisfiesy(x0)=0?
  2. Show that when x0≠0equation (13) can’t possibly have a solution in a neighbourhood of x=x0that satisfies y(x0)=0.
  3. Show two distinct solutions to (13) satisfying y(0)=0 ( See Figure 1.4 on page 9).

In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.

dydx=y4-x4,y(0)=7
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