1061
Concepts and Vocabulary
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 \(\dfrac{dy}{dx}+3=8e^{5x}.\)
Skill Building
In Problems 5–14, state the order and degree of each equation. State whether the equation is linear or nonlinear.
\(\dfrac{dy}{dx}+x^{2}y=xe^{x}\)
\(\dfrac{d^{3}y}{dx^{3}}+4\dfrac{d^{2}y}{dx^{2}}-5\dfrac{dy}{dx} +3y=\sin x\)
\(\dfrac{d^{4}y}{dx^{4}}+3\dfrac{d^{2}y}{dx^{2}}+5y=0\)
\(\dfrac{d^{2}y}{dx^{2}}+y\sin x=0\)
\(\dfrac{d^{2}y}{dx^{2}}+x\sin y=0\)
\(\dfrac{d^{6}x}{dt^{6}}+\dfrac{d^{4}x}{dt^{4}}+\dfrac{d^{3}x}{dt^{3}}+x=t\)
\(\left( {\dfrac{dr}{ds}}\right) ^{2}=\left(\dfrac{d^{2}r}{ds^{2}}\right) ^{3}+1\)
\(x( y^{\prime \prime}) ^{3}+ ( y^{\prime}) ^{4}-y=0\)
\(\dfrac{d^{2}y}{ds^{2}}+3s\dfrac{dy}{ds}=y\)
\(\dfrac{dy}{dx}=1-xy+y^{2}\)
In Problems 15–22, show that the function \(y\) is a solution of the differential equation.
\(y=e^{x}+3e^{-x},\quad \dfrac{d^{2}y}{dx^{2}}-y=0\)
\(y=5\sin x+2\cos x,\quad \dfrac{d^{2}y}{dx^{2}}+y=0\)
\(y=\sin ( 2x),\quad \dfrac{d^{2}y}{dx^{2}}+4y=0\)
\(y=\cos ( 2x) ,\quad \dfrac{d^{2}y}{dx^{2}}+4y=0\)
\(y=\dfrac{3}{1-x^{3}},\quad \dfrac{dy}{dx}=x^{2}y^{2}\)
\(y=1+Ce^{-x^{2}/2},\quad \dfrac{dy}{dx}+xy=x\), \(C\) is a constant
\(y=e^{-2x},\quad y^{\prime \prime} +3y^{\prime} +2y=0\)
\(y=e^{x},\quad 2y^{\prime \prime} +y^{\prime} -3y=0\)
In Problems 23–30, use implicit differentiation to show that the given equation is a solution of the differential equation. \(C\) is a constant.
\(y^{2}+2xy-x^{2}=C,\quad \dfrac{dy}{dx}=\dfrac{x-y}{x+y}\)
\(x^{2}-y^{2}=Cx^{2}y^{2},\quad \dfrac{dy}{dx}=\dfrac{y^{3}}{x^{3}}\)(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.)
Applications and Extensions
In Problems 31–34, for each differential equation, show that \(y\) is a particular solution that satisfies the initial condition.
\(\dfrac{dy}{dx}=\dfrac{y}{x};\quad y=3x\) 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 \(x=3\)
In Problems 35–38, show that \(y\) is a solution of the differential equation.
\(y=C_{1}e^{ax}+C_{2}e^{-ax},\quad \dfrac{d^{2}y}{dx^{2}} -a^{2}y=0\), \(C_1\), \(C_2\) 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?