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Chapter 8: Q-8-3-12P (page 403)

Using(3.9) , find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after(3.9) , and Example 1.

dxdy=cosy-xtany

Short Answer

Expert verified

The general solution of the differential equations isx=ycosy+Ccosy

Step by step solution

01

 Step 1: Given Information.

The given differential equations isdxdy=cosy-xtany

02

Step 2: Meaning of the first-order differential equation.

A first-order differential equation is defined by two variables,xandy, and its functionf(x,y)is defined on anXY-plane region.

03

Find the general solution.

Write this differential equation to make it in the formy'+Py=Q, that is

x'+xtany=cosy

From eq. 3.4,

I=鈭�tanydy=-ln(cosy)rI=1cosy

Find a solution for this differential equation

xeI=鈭�1cosycosydy=y+Cx=ycosy+Ccosy

Therefore, the general solution of the differential equations is x=ycosy+Ccosy.

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