Question

xdy/dx + (x+1) y = x³

Answer 1

$x\frac{dy}{dx}+(x+1)y=x^3$

$\frac{dy}{dx}+\frac{(x+1)}{x}y=x^2$

This is a linear differential equation of the first order

$I.F=e^{\int \frac{x+1}{x}}dx=e^{\int \frac{1}{x}+1dx}=e^{ln\ x+x}$

Solution to the equation;

$y\cdot (I.F)=\int x^2(I.F)dx+C$

$ye^{ln\ x+x}=\int x^2(e^{ln\ x+x})dx+C$

$y\cdot e^{ln\ x+x}=\int x^2\cdot e^{ln\ x}\cdot e^{x}dx+C=\int x^3\cdot e^{x}dx+C$

$y\cdot(x\cdot e^x)=[(x^3)(e^x)-(3x^2)(e^x)+(6x)(e^x)-(6)(e^x)]+C$

$y\cdot(x)=x^3-3x^2+6x-6+Ce^{-x}$

$y=x^2-3x+6-\frac{6}{x}+\frac{C}{x}e^{-x}$