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Chapter 1: Q14E (page 1)

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

If it is, then solve it.

(ty)dy+(1+lny)dt=0

Short Answer

Expert verified

The solution is tlny+t=C.

Step by step solution

01

Evaluate whether the equation is exact

Heretydy+(1+lny)dt=0

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

M(y,t)=tyN(y,t)=1+lny

鈭�M鈭�y=1y=鈭�N鈭�t

This equation is exact.

02

Find the value of F(y, t)

Here

M(y,t)=tyF(y,t)=鈭�M(y,t)dy+g(t)=鈭�tydy+g(t)=tlny+g(t)

03

determine the value of g (t)

NowFy,t= tlny + t +C1

The general solution of the differential equation is tlny+t=C.

Hence the solution istlny+t=C

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