Differential Equations — Practice

Question 1

Find the order and degree of $\left(\dfrac{d^{2}y}{dx^{2}}\right)^{3}+\sqrt{1+\dfrac{dy}{dx}}=0$.

Accepted Answer

Order is $2$, from the highest derivative.

Isolating and squaring gives $\left(\dfrac{d^{2}y}{dx^{2}}ight)^{6}=1+\dfrac{dy}{dx}$, which is polynomial in the derivatives.

Degree is $6$.

Question 2

Form the differential equation of the family $y=Ae^{2x}+Be^{-2x}$.

Accepted Answer

$y'=2Ae^{2x}-2Be^{-2x}$ and $y''=4Ae^{2x}+4Be^{-2x}$.

So $y''=4y$, that is $\dfrac{d^{2}y}{dx^{2}}-4y=0$.

Both constants vanish at the second differentiation, matching the count.

Question 3

Solve $\dfrac{dy}{dx}=y	an x$ completely, including any singular solution.

Accepted Answer

$\dfrac{dy}{y}=	an x\,dx$ gives $\ln|y|=\ln|\sec x|+C$, so $y=A\sec x$.

The division assumed $y
eq0$, and $y\equiv0$ does satisfy the equation.

Here it is recovered by $A=0$, so there is no singular solution — but the check was still necessary.

Question 4

Solve $\left(x^{2}-y^{2}\right)dx+2xy\,dy=0$.

Accepted Answer

$\dfrac{dy}{dx}=\dfrac{y^{2}-x^{2}}{2xy}$. Put $y=vx$:

$v+x\dfrac{dv}{dx}=\dfrac{v^{2}-1}{2v}$, so $x\dfrac{dv}{dx}=-\dfrac{1+v^{2}}{2v}$.

$\dfrac{2v\,dv}{1+v^{2}}=-\dfrac{dx}{x}$ gives $\ln\left(1+v^{2}ight)=-\ln|x|+C$.

So $x\left(1+\dfrac{y^{2}}{x^{2}}ight)=A$, that is $x^{2}+y^{2}=Ax$.

Question 5

Solve $\dfrac{dy}{dx}+2y	an x=\sin x$, given $y=0$ when $x=\dfrac{\pi}{3}$.

Accepted Answer

$\mu=e^{\int2	an x\,dx}=\sec^{2}x$.

$\dfrac{d}{dx}\left(y\sec^{2}xight)=\sin x\sec^{2}x=\sec x	an x$.

So $y\sec^{2}x=\sec x+C$, that is $y=\cos x+C\cos^{2}x$.

At $x=	frac{\pi}{3}$: $0=	frac12+	frac C4$, so $C=-2$ and $y=\cos x-2\cos^{2}x$.

Question 6

Solve $\left(1+y^{2}ight)dx=\left(	an^{-1}y-xight)dy$.

Accepted Answer

Rearranging, $\dfrac{dx}{dy}+\dfrac{x}{1+y^{2}}=\dfrac{	an^{-1}y}{1+y^{2}}$, linear in $x$.

$\mu=e^{	an^{-1}y}$, so $\dfrac{d}{dy}\left(xe^{	an^{-1}y}ight)=\dfrac{	an^{-1}y}{1+y^{2}}e^{	an^{-1}y}$.

With $u=	an^{-1}y$ the right side is $ue^{u}$, integrating to $(u-1)e^{u}$.

$x=	an^{-1}y-1+Ce^{-	an^{-1}y}$.

Question 7

Solve $\dfrac{dy}{dx}+\dfrac yx=x^{2}y^{6}$.

Accepted Answer

Dividing by $y^{6}$: $y^{-6}y'+\dfrac{y^{-5}}{x}=x^{2}$.

Put $v=y^{-5}$, so $v'=-5y^{-6}y'$ and the equation becomes $-	frac15v'+\dfrac vx=x^{2}$.

$v'-\dfrac{5v}{x}=-5x^{2}$, with $\mu=x^{-5}$.

$\dfrac{d}{dx}\left(vx^{-5}ight)=-5x^{-3}$, so $vx^{-5}=\dfrac{5}{2x^{2}}+C$ and $y^{-5}=\dfrac{5x^{3}}{2}+Cx^{5}$.

Question 8

Solve $x\,dy-y\,dx=xy\,dx$.

Accepted Answer

Divide by $xy$: $\dfrac{x\,dy-y\,dx}{xy}=dx$.

The left side is $d\!\left(\ln\dfrac yxight)$.

Integrating, $\ln\dfrac yx=x+C$, so $y=Axe^{x}$.

Question 9

Find the curve through $(1,2)$ whose subnormal at every point is $4$.

Accepted Answer

$y\dfrac{dy}{dx}=4$, so $\dfrac{y^{2}}{2}=4x+C$.

At $(1,2)$: $2=4+C$, so $C=-2$.

$y^{2}=8x-4$, a parabola.

Question 10

Find the orthogonal trajectories of the family $x^{2}+y^{2}=2ax$.

Accepted Answer

Differentiating, $2x+2yy'=2a$, so $a=x+yy'$.

Substituting into $x^{2}+y^{2}=2ax$ gives $y^{2}-x^{2}=2xyy'$.

Replacing $y'$ by $-\dfrac{1}{y'}$: $y^{2}-x^{2}=-\dfrac{2xy}{y'}$, that is $\dfrac{dy}{dx}=\dfrac{2xy}{x^{2}-y^{2}}$.

Solving as a homogeneous equation gives $x^{2}+y^{2}=Cy$: circles through the origin centred on the other axis.

Question 11

A radioactive substance decays so that $10\%$ is lost in $2$ years. Find its half-life.

Accepted Answer

$N=N_0e^{-kt}$ with $e^{-2k}=0.9$.

Half-life $T$ satisfies $e^{-kT}=	frac12$.

Taking logarithms, $\dfrac{T}{2}=\dfrac{\ln 2}{\ln(1/0.9)}$, so $T=\dfrac{2\ln2}{\ln(10/9)}\approx13.2$ years.

Question 12

Water leaks from a tank so that the depth $h$ falls at a rate proportional to $\sqrt h$. The depth falls from $16$ m to $9$ m in one hour. When is the tank empty?

Accepted Answer

$\dfrac{dh}{dt}=-k\sqrt h$ gives $2\sqrt h=-kt+C$.

At $t=0$, $h=16$: $C=8$. At $t=1$, $h=9$: $6=-k+8$, so $k=2$.

Empty when $h=0$: $0=-2t+8$, so $t=4$ hours.

Practice — Differential Equations

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