The differential equation $\dfrac{d^{2}y}{dx^{2}}+5\left( \dfrac{dy}{dx}\right) ^{3}-y=0$ is of order _________ and degree ________.

True or False The differential equation $(x^{2}+2) \dfrac{dy}{dx}-\dfrac{1}{x}y=0$ is linear.

True or False $y=-4x^{2}+100$ is a solution of the differential equation $\dfrac{dy}{dx}=-8x.$

True or False $y=8e^{5x}$ is a solution of the differential equation $\dfrac{dy}{dx}+3=8e^{5x}.$

$\dfrac{dy}{dx}+x^{2}y=xe^{x}$

$\dfrac{d^{3}y}{dx^{3}}+4\dfrac{d^{2}y}{dx^{2}}-5\dfrac{dy}{dx} +3y=\sin x$

$\dfrac{d^{4}y}{dx^{4}}+3\dfrac{d^{2}y}{dx^{2}}+5y=0$

$\dfrac{d^{2}y}{dx^{2}}+y\sin x=0$

$\dfrac{d^{2}y}{dx^{2}}+x\sin y=0$

$\dfrac{d^{6}x}{dt^{6}}+\dfrac{d^{4}x}{dt^{4}}+\dfrac{d^{3}x}{dt^{3}}+x=t$

$\left( {\dfrac{dr}{ds}}\right) ^{2}=\left(\dfrac{d^{2}r}{ds^{2}}\right) ^{3}+1$

$x( y^{\prime \prime}) ^{3}+ ( y^{\prime}) ^{4}-y=0$

$\dfrac{d^{2}y}{ds^{2}}+3s\dfrac{dy}{ds}=y$

$\dfrac{dy}{dx}=1-xy+y^{2}$

$y=e^{x}+3e^{-x},\quad \dfrac{d^{2}y}{dx^{2}}-y=0$

$y=5\sin x+2\cos x,\quad \dfrac{d^{2}y}{dx^{2}}+y=0$

$y=\sin ( 2x),\quad \dfrac{d^{2}y}{dx^{2}}+4y=0$

$y=\cos ( 2x) ,\quad \dfrac{d^{2}y}{dx^{2}}+4y=0$

$y=\dfrac{3}{1-x^{3}},\quad \dfrac{dy}{dx}=x^{2}y^{2}$

$y=1+Ce^{-x^{2}/2},\quad \dfrac{dy}{dx}+xy=x$, $C$ is a constant

$y=e^{-2x},\quad y^{\prime \prime} +3y^{\prime} +2y=0$

$y=e^{x},\quad 2y^{\prime \prime} +y^{\prime} -3y=0$

$y^{2}+2xy-x^{2}=C,\quad \dfrac{dy}{dx}=\dfrac{x-y}{x+y}$

$x^{2}-y^{2}=Cx^{2}y^{2},\quad \dfrac{dy}{dx}=\dfrac{y^{3}}{x^{3}}$(Hint: Solve for $C$ first.)

$x^{2}-y^{2}=Cx,\quad \dfrac{dy}{dx}=\dfrac{x^{2}+y^{2}}{2xy}$ (Hint: Solve for $C$ first.)

$x^{2}y+12x^{2}+16y=9,\quad \dfrac{dy}{dx}=-\dfrac{2xy+24x}{x^{2}+16}$

$2\left( x-1\right) e^{x}+y^{2}=C,\dfrac{dy}{dx}=-\dfrac{xe^{x}}{y}$

$x^{2}-y^{2}+2( e^{-x}-e^{y}) =C,\quad \dfrac{dy}{dx}=\dfrac{x-e^{-x}}{y+e^{y}}$

$\tan ^{-1}y=x+\dfrac{x^{2}}{2}+C,\quad \dfrac{dy}{dx}=xy^{2}+y^{2}+x+1$

$1+2y^{2}\sqrt{1+x^{2}}=Cy^{2},\quad \dfrac{dy}{dx}=\dfrac{xy^{3}}{\sqrt{1+x^{2}}}$ (Hint: Solve for $C$ first.)

$\dfrac{dy}{dx}=\dfrac{y}{x};\quad y=3x$ initial condition: $y=3$ when $x=1$

$\dfrac{dy}{dx}=3y;\quad y=2e^{3x}$ initial condition: $y=2$ when $x=0$

$\dfrac{dy}{dx}=y^{2};\quad y=\dfrac{1}{1-x}$ initial condition: $y=1$ when $x=0$

$x\dfrac{dy}{dx}+y=x^{2};\quad y=\dfrac{x^{2}}{3}+\dfrac{3}{x}$ initial condition: $y=4$ when $x=3$

$y=C_{1}e^{ax}+C_{2}e^{-ax},\quad \dfrac{d^{2}y}{dx^{2}} -a^{2}y=0$, $C_1$, $C_2$ are constants

$y=\sinh x,\quad \dfrac{d^{2}y}{dx^{2}}=y$

$y=\dfrac{a^{2}kt}{1+akt},\quad \dfrac{dy}{dt}=k(a-y)^{2}$, $a>0$, and $k$ are constants

$y=\ln (C-e^{-x}),\quad \dfrac{dy}{dx}=e^{-(x+y)}$, $C$ is a constant

Find the values of $n$ so that $y=e^{nx}$ is a solution of $$y^{\prime \prime} +y^{\prime} -6y=0$$

Find a first-order differential equation that has $y=e^{x}+e^{-x}$ as a solution.

Schrödinger Equation In quantum mechanics, the time-independent Schrödinger equation in one dimension can be written as $-\dfrac{h^{2}}{2m}\dfrac{d^{2}Y( x) }{dx^{2}}+U( x) Y( x) =EY( x) .$ What are the degree and order of this differential equation?