Chapter 30

Solve each differential equation. $$3 D y=5 x+2$$

Find the general solution of each differential equation. Try some by calculator. $$(2 x-y) y^{\prime}=x-2 y$$

Find the general solution to each differential equation. $$y^{\prime}=\frac{2-4 x^{2} y}{x+x^{3}}$$

An instrument package is dropped from an airplane. It falls from rest through air whose resisting force is proportional to the speed of the package. The terminal speed is \(155 \mathrm{ft} / \mathrm{s}\). Show that the acceleration is given by the differential equation \(a=d v / d t=g-g v / 155\)

Solve each differential equation. $$d y=x^{2} d x$$

Find the general solution to each differential equation. $$(x+1) y^{\prime}=2(x+y+1)$$

Solve each differential equation. $$d y-4 x d x=0$$

Find the general solution to each differential equation. $$x y^{\prime}+x^{2} y+y=0$$

Show that each function is a solution to the given differential equation. $$y^{\prime}=\frac{2 y}{x}, y=C x^{2}$$

Find the general solution of each differential equation. Try some by calculator. $$\left(x-2 x^{2} y\right) d y=y d x$$

Find the general solution to each differential equation. $$\left(1+x^{3}\right) d y=\left(1-3 x^{2} y\right) d x$$

Show that each function is a solution to the given differential equation. $$\frac{d y}{d x}=\frac{x^{2}}{y^{3}}, 4 x^{3}-3 y^{4}=C$$

For the circuit of Fig. \(30-13, R=233 \Omega, L=5.82 \mathrm{H},\) and \(E=58.0 \mathrm{sin} 377 t \mathrm{V}\). If the switch is closed when \(E\) is zero and increasing, show that the current is given by \(i=26.3 \sin \left(377 t-83.9^{\circ}\right)+26.1 e^{-40 t} \mathrm{mA}.\)

Find the general solution to each differential equation. $$y^{\prime}+y=e^{x}$$

Show that each function is a solution to the given differential equation. $$D y=\frac{2 y}{x}, y=C x^{2}$$

Find the general solution of each differential equation. Try some by calculator. $$(y-x) y^{\prime}+2 x y^{2}+y=0$$

Find the general solution to each differential equation. Try some by calculator. $$y^{\prime}=\frac{e^{x-y}}{e^{x}+1}$$

Show that each function is a solution to the given differential equation. $$y^{\prime} \cot x+3+y=0, y=C \cos x-3$$

Find the general solution of each differential equation. Try some by calculator. $$3 x-2 y^{2}-4 x y y^{\prime}=0$$

Find the general solution to each differential equation. $$y^{\prime}=2 y+4 e^{2 x}$$

Find the general solution to each differential equation. $$x y^{\prime}-e^{x}+y+x y=0$$

A box falls from rest and encounters air resistance proportional to the cube of the speed. The limiting speed is \(12.5 \mathrm{ft} / \mathrm{s}\). Show that the acceleration is given by the differential equation \(60.7 d v / d t=1953-v^{3}\)

With Exponential Functions $$d y=e^{-x} d x$$

Using the given boundary condition, find the particular solution to each differential equation. $$4 x=y+x y^{\prime}, x=3 \text { when } y=1$$

Find the general solution to each differential equation. $$y^{\prime}=\frac{4 \ln x-2 x^{2} y}{x^{3}}$$

With Exponential Functions $$y e^{2 x}=\left(1+e^{2 x}\right) y^{\prime}$$

Using the given boundary condition, find the particular solution to each differential equation. $$y d x=\left(x-2 x^{2} y\right) d y, x=1 \text { when } y=2$$

$$y^{\prime}+y \sin x=3 \sin x$$

With Exponential Functions $$e^{y}\left(y^{\prime}+1\right)=1$$

Using the given boundary condition, find the particular solution to each differential equation. $$y=\left(3 y^{3}+x\right) \frac{d y}{d x}, x=1 \text { when } y=1$$

$$y^{\prime}+y=\sin x$$

With Exponential Functions $$e^{x-y} d x+e^{y-x} d y=0$$

Using the given boundary condition, find the particular solution to each differential equation. $$4 x^{2}=-2 y-2 x y^{\prime}, x=5 \text { when } y=2$$

$$y^{\prime}+2 x y=2 x \cos x^{2}$$

Using the given boundary condition, find the particular solution to each differential equation. $$4 x^{2}=-2 y-2 x y^{\prime}, x=5 \text { when } y=2$$

$$y^{\prime}=2 \cos x-y$$

Using the given boundary condition, find the particular solution to each differential equation. $$5 x=2 y^{2}+4 x y y^{\prime}, x=1 \text { when } y=4$$

$$y^{\prime}=\sec x-y \cot x$$

With Trigonometric Functions $$\tan y d x+\tan x d y=0$$

Bernoulli's Equation. $$y^{\prime}+\frac{y}{x}=3 x^{2} y^{2}$$

With Trigonometric Functions $$\cos x \sin y d y+\sin x \cos y d x=0$$

Bernoulli's Equation. $$x y^{\prime}+x^{2} y^{2}+y=0$$

Bernoulli's Equation. $$y^{\prime}=y-x y^{2}(x+2)$$

Bernoulli's Equation. $$y^{\prime}+2 x y=x e^{-x^{2}} y^{3}$$

Using the given boundary condition, find the particular solution to each differential equation. Try some by calculator. $$x d x=4 y d y, x=5 \text { when } y=2$$

Using the given boundary condition, find the particular solution to each differential equation. Try some by calculator. $$y^{2} y^{\prime}=x^{2}, x=0 \text { when } y=1$$

Using the given boundary condition, find the particular solution to each differential equation. Try some by calculator. $$\sqrt{x^{2}+1} y^{\prime}+3 x y^{2}=0, x=1 \text { when } y=1$$

Using the given boundary condition, find the particular solution to each differential equation. Try some by calculator. $$y^{\prime} \sin y=\cos x, x=\pi / 2 \text { when } y=0$$

Using the given boundary condition, find the particular solution to each differential equation. Try some by calculator. $$x(y+1) y^{\prime}=y(1+x), x=1 \text { when } y=1$$

Using the given boundary condition, find the particular solution to each differential equation. $$y^{\prime}=\tan ^{2} x+y \cot x, x=\frac{\pi}{4} \text { when } y=2$$