A$.$ Solve the given first order differential equations.

$x^2+y(x-1)y^{\text{'}}=0$

Ans: $(x+1{)}^2+y^2+2ln|c(x-1)|=0$

$2.(1+lnx)+(1+lny)y^{\text{'}}=0$

Ans: $xlnx+ylny=c$

$3.(x-y)+(3x+y)y^{\text{'}}=0$;$,x(3)=-2$

Ans: $2(2x+3y)+(x+y)ln(x+y)=0$

$4.(cosxcosy-cotx)-(sinxsiny)y^{\text{'}}=0$

Ans. $sinxcosy=ln|c(sinx)|$

$5.x^{2}y{y}^{\text{'}}=e^y$

$6.1=(1+2xtany)y^{\text{'}}$

Ans $2xco{s}^{2}y=y+c+sinycosy$

$7.(1+y^2)+(x^{2}y+y)y^{\text{'}}=0$

Ans: $2ta{n}^{-1}x+ln(1+y^2)=c$

$8.(x^5+3y)-x{y}^{\text{'}}=0$

Ans: $2y=x^5+c{x}^3$

$9.(y^2+2xy)-x^2{y}^{\text{'}}=0$

Ans: $x^2{y}^{-1}+x=c$

$10.(x^2+y^2+1)+x(x-2y)y^{\text{'}}=0$

Ans: $x^2-y^2+xy-1=cx$

$11.y^{\text{'}}=1+6x*e^{x-y}$

Ans: $e^{y-x}=3x^2+c$

$12.xco{s}^{2}y+tany*y^{\text{'}}=0$

Ans: $x^2+ta{n}^{2}y=c$

B$.$ Solve $(x+1{)}^2{y}^{\text{'}\text{'}}-2(x+1)y^{\text{'}}+2y=0$ by finding one solution by trial and error, and the second solution through reduction of order.

C$.$ Solve the given differential equations.

$y^{\text{'}\text{'}}+3y^{\text{'}}+2y=8+6e^x+2sinx$

Ans: $y=A{e}^{-2x}+B{e}^{-x}+4+e^x+\frac{1}{5}(sinx-3cosx)$

$2.y^{\text{'}\text{'}}+3y^{\text{'}}+2y=e^{-2x}+x^2$

Ans: $y=A{e}^{-2x}+B{e}^{-x}+\frac{7}{4}-\frac{3x}{2}+\frac{x^2}{2}-x{e}^{-2x}$

$3.y^{\text{'}\text{'}}-3y^{\text{'}\text{'}}+3y^{\text{'}}-y=e^x$

Ans: $y=(A+Bx+C{x}^2+\frac{x^3}{6})e^x$