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Chapter 2: Q15E (page 64)

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

If it is, then solve it.

cos胃dr-(rsin胃-e胃)d胃=0

Short Answer

Expert verified

The solution is r=C-e胃sec胃

Step by step solution

01

Evaluate whether the equation is exact

Herecos胃dr-(rsin胃-e胃)d胃=0

The condition for exact is鈭�M鈭�胃=鈭�N鈭�r.

M(r,胃)=cos胃N(r,胃)=-rsin胃+e胃

鈭�M鈭�胃=-sin胃=鈭�N鈭�r

This equation is exact.

02

Find the value of  F(r,θ)

Here,

M(r,胃)=cos胃F(r,胃)=鈭�M(r,胃)dr+g(胃)=鈭�cos胃dr+g(胃)=rcos胃+g(t)

03

Determine the value of g(θ)

鈭�F鈭�胃(r,胃)=N(r,胃)-rsin胃+g'(胃)=-rsin胃+e胃g'(y)=e胃g(y)=e胃

NowF(r,胃)=rcos胃+e胃

Therefore, the general solution of the differential equation is

rcos胃+e胃=Cr=C-e胃cos胃r=C-e胃sec胃

Hence the solution isrole="math" localid="1664173591620" r=(C-e胃)sec胃

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