91Ó°ÊÓ

Solve the given differential equations. $$\tan \theta \frac{d r}{d \theta}-r=\tan ^{2} \theta$$

Find the inverse transforms of the given functions of \(s\) $$F(s)=\frac{2 s+3}{s^{2}-2 s+5}$$

Solve the given equations by letting \(u=y / x\). $$d y=\left(\frac{y}{x}+\frac{y^{2}}{x^{2}}\right) d x$$

Solve the given differential equations. $$y^{\prime}-y=4$$

Show that the given equation is a solution of the given differential equation. $$\frac{d^{3} y}{d x^{3}}=\frac{d^{2} y}{d x^{2}}, \quad y=c_{1}+c_{2} x+c_{3} e^{x}$$

Solve the given problems by solving the appropriate differential equation. According to Newton's law of cooling, the rate at which a body cools is proportional to the difference in temperature between it and the surrounding medium. Assuming Newton's law holds, how long will it take a cup of hot water, initially at \(200^{\circ} \mathrm{F}\), to cool to \(100^{\circ} \mathrm{F}\) if the room temperature is \(80.0^{\circ} \mathrm{F},\) if it cools to \(140^{\circ} \mathrm{F}\) in \(5.0 \mathrm{min} ?\)

Show that the given equation is a solution of the given differential equation. $$2 x y y^{\prime}+x^{2}=y^{2}, \quad x^{2}+y^{2}=c x$$

Solve the given differential equations. $$36 D^{2} y=25 y$$

Solve the given equations by letting \(u=y / x\). $$d y=\left(\frac{x-y}{x}\right) d x$$

Solve the given differential equations. \(y^{\prime}+y=y^{2}\) (Solve by letting \(y=1 / u\) and solving the resulting linear equation for \(u\).)