91Ó°ÊÓ

In Problems \(41-50\) solve the given differential equation subject to the indicated initial condition. $$ y^{\prime}+(\tan x) y=\cos ^{2} x, \quad y(0)=-1 $$

In Problems \(41-50\) solve the given differential equation subject to the indicated initial condition. $$ \frac{d Q}{d x}=5 x^{4} Q, \quad Q(0)=-7 $$

Solve the given differential equation subject to the indicated initial condition. $$ \frac{d y}{d x}=\frac{y^{2}-1}{x^{2}-1}, \quad y(2)=2 $$

Solve the given differential equation subject to the indicated initial condition. $$ x^{2} y^{\prime}=y-x y, \quad y(-1)=-1 $$

In Problems \(41-50\) solve the given differential equation subject to the indicated initial condition. $$ \frac{d T}{d t}=k(T-50) ; \quad k \text { a constant, } T(0)=200 $$

In Problems \(41-50\) solve the given differential equation subject to the indicated initial condition. $$ x d y+\left(x y+2 y-2 e^{-x}\right) d x=0, \quad y(1)=0 $$

Solve the given differential equation subject to the indicated initial condition. $$ y^{\prime}+2 y=1, \quad y(0)=\frac{1}{2} $$

In Problems \(41-50\) solve the given differential equation subject to the indicated initial condition. $$ (x+1) \frac{d y}{d x}+y=\ln x, \quad y(1)=10 $$

find a solution of the given differential equation that passes through the indicated points. $$ \frac{d y}{d x}-y^{2}=-9 $$ (a) \((0,0)\) (b) \((0,3)\) (c) \(\left(\frac{1}{3}, 1\right)\)

Find a solution of the given differential equation that passes through the indicated points. $$ x \frac{d y}{d x}=y^{2}-y $$ (a) \((0,1)\) (b) \((0,0)\) (c) \(\left(\frac{1}{2}, \frac{1}{2}\right)\)