\((x+1)y'−3y=(x+1)^5\)

\(y'−3/((x+1))y=(x+1)^4\)

Compare with

\(y'+P(x)y= Q(x)\)

\(P(x)= -3/(x+1)\)

\(Q(x)= (x+1)^4\)

Hence given differential equation is of the linear type

\(I.F.= e^(int P(x)dx)\)

\(= e^(int -3/(x+1)dx)\(

\(= e^(-3 ln(x+1))\)

\(= e^(ln (x+1)^-3)\)

\(= (x+1)^-3\)

\(y*I.F.= int Q(x)*I.F. dx\)

\(y(x+1)^3= int (x+1)^4*(x+1)^-3 dx\)

\(y(x+1)^-3= int (x+1)dx\)

\(y(x+1)^-3= x^2/2+x+C\)

\(y=(x^2/2+x+C)(x+1)^3\)

\(y(0)=1.5\)

\((0+0+C)(0+1)^3=1.5\)

C=1.5

\(y=(x^2/2+x+1.5)(x+1)^3\)

\(y'−3/((x+1))y=(x+1)^4\)

Compare with

\(y'+P(x)y= Q(x)\)

\(P(x)= -3/(x+1)\)

\(Q(x)= (x+1)^4\)

Hence given differential equation is of the linear type

\(I.F.= e^(int P(x)dx)\)

\(= e^(int -3/(x+1)dx)\(

\(= e^(-3 ln(x+1))\)

\(= e^(ln (x+1)^-3)\)

\(= (x+1)^-3\)

\(y*I.F.= int Q(x)*I.F. dx\)

\(y(x+1)^3= int (x+1)^4*(x+1)^-3 dx\)

\(y(x+1)^-3= int (x+1)dx\)

\(y(x+1)^-3= x^2/2+x+C\)

\(y=(x^2/2+x+C)(x+1)^3\)

\(y(0)=1.5\)

\((0+0+C)(0+1)^3=1.5\)

C=1.5

\(y=(x^2/2+x+1.5)(x+1)^3\)