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Chapter 6: Q. 48 (page 570)

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29-52.

dydx=xy+x+y+1,y(0)=c

Short Answer

Expert verified

On solving, we gety(x)=-1+(1+c)e12(x+1)2

Step by step solution

01

Step 1. Given information

Given the expressiondydx=xy+x+y+1,y(0)=c

02

Take common and use variable separable method

Calculating, we get

dydx=x(y+1)+1(y+1)=(x+1)(y+1)

Integrating, we get

鈭�1y+1dy=鈭�(x+1)dxln|y+1|=12(x+1)2+Cy+1=e12(x+1)2+Cy=-1+Ae12(x+1)2

03

Substitute x=0,y=c in the equation and solve

Calculating, we get

c=-1+AA=1+cy(x)=-1+(1+c)e12(x+1)2

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