Question

<e> Solve the first order linear inhomogeneous differential equation using the bernoulli method

(2x+1)y^,=4x+2y

Answer 1

$(2x+1)y'=4x+2y\\ y=u(x)v(x)\\ (2x+1)(u'v+uv')=4x+2uv\\ (2x+1)u'v+(2x+1)uv'-2uv=4x\\ (2x+1)u'v+u((2x+1)v'-2v)=4x\\ 1) (2x+1)v'-2v=0\\ \frac{dv}{v}=\frac{2dx}{2x+1}\\ \int\frac{dv}{v}=\int\frac{2dx}{2x+1}\\ \ln|v|=\ln|2x+1|\\ v=2x+1\\ 2) (2x+1)u'\cdot (2x+1)=4x\\ du=\frac{4x}{(2x+1)^2}\\ u=\int\frac{4x}{(2x+1)^2}dx=\\ =\int\frac{2(2x+1-1)}{(2x+1)^2}dx=\\ =\int(\frac{2}{2x+1}-\frac{2}{(2x+1)^2})dx=\\ =\ln|2x+1|+\frac{1}{2x+1}+C\\ y= (2x+1)(\ln|2x+1|+\frac{1}{2x+1}+C)$