Section 1
Solutions of Differential Equations
Show that the indicated solutions are, in fact, solutions of the differential equations in the indicated examples.In Example $3,$ two solutions are shown for the given differential equation. Show that each is a solution.
Determine whether the given equation is the general solution or a particular solution of the given differential equation.$$\frac{d y}{d x}+2 x y=0, \quad y=e^{-x^{2}}$$
Determine whether the given equation is the general solution or a particular solution of the given differential equation.$$y^{\prime} \ln x-\frac{y}{x}=0, \quad y=c \ln x$$
Determine whether the given equation is the general solution or a particular solution of the given differential equation.$$y^{\prime \prime}+3 y^{\prime}-4 y=3 e^{x}, \quad y=c_{1} e^{x}+c_{2} e^{-4 x}+\frac{3}{5} x e^{x}$$
Determine whether the given equation is the general solution or a particular solution of the given differential equation.$$\frac{d^{2} y}{d x^{2}}+4 y=8, \quad y=c \sin 2 x+3 \cos 2 x+2$$
Show that each function $y=f(x)$ is a solution of the given differential equation.$$\frac{d y}{d x}-y=1 ; \quad y=e^{x}-1, \quad y=5 e^{x}-1$$
Show that each function $y=f(x)$ is a solution of the given differential equation.$$\frac{d y}{d x}=2 x y^{2} ; \quad y=-\frac{1}{x^{2}}, \quad y=-\frac{1}{x^{2}+c}$$
Show that each function $y=f(x)$ is a solution of the given differential equation.$$y^{\prime \prime}+4 y=0 ; \quad y=3 \cos 2 x, \quad y=c_{1} \sin 2 x+c_{2} \cos 2 x$$
Show that each function $y=f(x)$ is a solution of the given differential equation.$$y^{\prime \prime}=2 y^{\prime} ; \quad y=3 e^{2 x}, \quad y=2 e^{2 x}-5$$
Show that the given equation is a solution of the given differential equation.$$\frac{d y}{d x}=2 x, \quad y=x^{2}+1$$
Show that the given equation is a solution of the given differential equation.$$x y^{\prime}=2 y, \quad y=c x^{2}$$
Show that the given equation is a solution of the given differential equation.$$\frac{d y}{d x}=1-3 x^{2}, \quad y=2+x-x^{3}$$
Show that the given equation is a solution of the given differential equation.$$\frac{d y}{d x}=3 y+2 x, \quad y=c e^{3 x}-\frac{2}{3} x-\frac{2}{9}$$
Show that the given equation is a solution of the given differential equation.$$y^{\prime}+2 y=2 x, \quad y=c e^{-2 x}+x-\frac{1}{2}$$
Show that the given equation is a solution of the given differential equation.$$y^{\prime \prime}=6 x+2, \quad y=x^{3}+x^{2}+c$$
Show that the given equation is a solution of the given differential equation.$$y^{\prime \prime}+9 y=4 \cos x, \quad 2 y=\cos x$$
Show that the given equation is a solution of the given differential equation.$$y^{\prime}=y \sec x, \quad y=\sec x+\tan x$$
Show that the given equation is a solution of the given differential equation.$$x^{2} y^{\prime}+y^{2}=0, \quad x y=c x+c y$$
Show that the given equation is a solution of the given differential equation.$$x y^{\prime}-3 y=x^{2}, \quad y=c x^{3}-x^{2}$$
Show that the given equation is a solution of the given differential equation.$$x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}=0, \quad y=c_{1} \ln x+c_{2}$$
Show that the given equation is a solution of the given differential equation.$$y^{\prime \prime}+4 y=10 e^{x}, \quad y=c_{1} \sin 2 x+c_{2} \cos 2 x+2 e^{x}$$
Show that the given equation is a solution of the given differential equation.$$y^{\prime}+y=2 \cos x, \quad y=\sin x+\cos x-e^{-x}$$
Show that the given equation is a solution of the given differential equation.$$(x+y)-x y^{\prime}=0, \quad y=x \ln x-c x$$
Show that the given equation is a solution of the given differential equation.$$y^{\prime \prime}-3 y^{\prime}+2 y=3, \quad y=c_{1} e^{x}+c_{2} e^{2 x}+3 / 2$$
Show that the given equation is a solution of the given differential equation.$$x y^{\prime \prime}+y^{\prime}=16 x^{3}, \quad y=x^{4}+c_{1}+c_{2} \ln x$$
Show that the given equation is a solution of the given differential equation.$$\frac{d^{3} y}{d x^{3}}=\frac{d^{2} y}{d x^{2}}, \quad y=c_{1}+c_{2} x+c_{3} e^{x}$$
Show that the given equation is a solution of the given differential equation.$$2 x y y^{\prime}+x^{2}=y^{2}, \quad x^{2}+y^{2}=c x$$
Show that the given equation is a solution of the given differential equation.$$\left(y^{\prime}\right)^{2}+x y^{\prime}=y, \quad y=c x+c^{2}$$
Show that the given equation is a solution of the given differential equation.$$x^{4}\left(y^{\prime}\right)^{2}-x y^{\prime}=y, \quad y=c^{2}+\frac{c}{x}$$
Determine whether or not each of the given functions is a solution of the differential equation $y^{\prime \prime}-2 y^{\prime}-3 y=-4 e^{x}$$$y=e^{x}+e^{-x}$$
Determine whether or not each of the given functions is a solution of the differential equation $y^{\prime \prime}-2 y^{\prime}-3 y=-4 e^{x}$$$y=e^{3 x}+e^{-x}$$
Determine whether or not each of the given functions is a solution of the differential equation $y^{\prime \prime}-2 y^{\prime}-3 y=-4 e^{x}$$$y=e^{x}+e^{2 x}$$
Determine whether or not each of the given functions is a solution of the differential equation $y^{\prime \prime}-2 y^{\prime}-3 y=-4 e^{x}$$$y=2 e^{-x}+e^{x}$$
Solve the given problems.The general solution of the differential equation $y^{\prime \prime}-y^{\prime} / x=3 x$ is $y=x^{3}+c_{1} x^{2}+c_{2} .$ Find the particular solution if the graph of the solution passes through the point (0,-4)
Solve the given problems.Find the particular solution of the differential equation in Exercise 35 if the graph of the solution passes through (0,-4) and (2,8)
Solve the given problems.A differential equation that arises in the study of radioactivity is $d N / d t=k N .$ Show that $N=N_{0} e^{t^{t}}$ is the general solution.
Solve the given problems.Show that the electric charge $q=0.01(1-\cos 316 t)$ in a circuit, where $t$ represents time satisfies the equation $\frac{d^{2} q}{d t^{2}}+10^{5} q=10^{3}$