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Chapter 2: Q10E (page 64)

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

If it is, then solve it.

(2x+y)dx+(x-2y)dy=0

Short Answer

Expert verified

The solution is x2+ yx -y2= C

Step by step solution

01

Evaluate whether the equation is exact

Here (2x+y)dx+(x-2y)dy=0

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

M(x,y)=2x+yN(x,y)=x-2y鈭�M鈭�y=1=1=鈭�N鈭�x

This equation is exact.

02

Find the value of F (x, y)

Here

M(x,y)=2x+yF(x,y)=鈭�M(x,y)dx+g(y)=鈭�(2x+y)dx+g(y)=x2+yx+g(y)

03

Determine the value of g(y)

鈭�F鈭�y(x,y)=N(x,y)x+g'(y)=x-2yg'(y)=-2yg(y)=-y2+C1

NowF(x,y)=x2+yx-y2+C1

The general solution of the differential equation isx2+ yx -y2= C.

Therefore, the solution isrole="math" localid="1664170796558" x2+ yx -y2=C

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Most popular questions from this chapter

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