Solve the following initial value problem,

$y^{\text{'}}(xy+y-2x-2)=ln(x+1),y(0)=0,x>-1.$

Find a special integrating factor and solve

$xy{y}^{\text{'}}+x+y^2=0$

Find $an$ integrating factor and solve

$(2x^{2}y+x)dy+(x{y}^2+y)dx=0$

Solve the following initial value problem,

$xdy+(y-y^{2}lnx)dx=0,y(1)=\frac{1}{4}$

Solve

$2x\frac{dy}{dx}-y=y[1-l{n}^2(\frac{y}{x})],x>0$

Solve

$\frac{dy}{dx}-co{s}^2(x-y)=0$

Solve

$(y^{\text{'}}{)}^2+(x+2y)cos(x+y)=(x+2y+cos(x+y))y^{\text{'}}$

A tank $is$ filled with $V=200Lof$ a brine containing $=\mathrm{.}4kgof$ salt A per litre. $At$ moment $0,$ input

and output valves are opened, and a brine containing another salt $B,$ with concentration $=\mathrm{.}2kg$ per

litre runs into the tank $at$ a rate $r_i=5\frac{L}{sec}.$ The mix runs out $of$ the tank with rate $r_o=4\frac{L}{sec}.$ The

salts $do$ not interact with each other. Determine the ratio $kof$ quantity $of$ salt $Bto$ the quantity $of$ salt $A$

when the tank contains $V_1=400Lof$ the mixture.