Differential Equations (DP IB Applications & Interpretation (AI): HL): Exam Questions

Consider the first-order differential equation

 $\frac{dy}{dx}-x^3=2\sin x$ 

Solve the equation given that $y=0$ when $x=0$,  giving your answer in the form $y=f\left(x\right)$.

Use separation of variables to solve each of the following differential equations:

$\frac{dy}{dx}=10x^3{y}^3$

$\frac{dy}{dx}=x{\left(x^2-1\right)}^3{e}^{3y}$

Use separation of variables to solve each of the following differential equations for $y$ which satisfies the given boundary condition:

$\frac{dy}{dx}=\frac{\cos 3x}{y};y\left(\frac{\mathrm{\pi }}{6}\right)=-1$

$e^{2x}\frac{dy}{dx}={\cos }^{2}y;y\left(0\right)=\frac{\mathrm{\pi }}{4}$

Radiangast the Beige is chief mathemagician of the wizards’ council.  After animals begin falling ill in the forest where he lives, Radiangast realises that an evil magic has begun spreading through the forest.  After studying the situation, he believes that at any point in time, $t$, the rate of change of the area, $A$, affected by the evil magic is inversely proportional to the square root of the area already affected. 

Write down a differential equation representing Radiangast’s model, and solve it to find the general solution.  Be sure to define any constants that occur in your equation or solution.

At the time when Radiangast first noticed its presence, the evil magic was affecting an area of 16 acres of forest.  One week later he noticed that the area has increased to 41 acres.

Radiangast knows that as long as the wizards’ council convenes to weave spells before the area affected by the evil magic exceeds 100 acres, then they will be able to stop the evil magic from spreading further. 

From the time that Radiangast first noticed the presence of the evil magic, determine how long the wizards’ council has to convene to weave spells, if they are to stop the evil magic from spreading further.

After an invasive species of insect has been introduced to a new region, it is estimated that at any point in time  the rate of growth of the population of insects in the region will be proportional to the current population size $P$. At the start of a study of the insects in a particular region, researchers estimate the population size to be 1000 individuals. A week later another population survey is conducted, and the population of insects is found to have increased to 1150.

By first writing and solving an appropriate differential equation, determine how long it will take for the population of insects in the region to increase to 10 000.

Comment on the validity of the model for large values of $t$.

The graph below shows the slope field for the differential equation $\frac{dy}{dx}=\frac{\left(0.2x-0.85\right)y}{\left(0.75-0.2y\right)x},$ $x>0,y>0,$ in the intervals $0<x\le 10$ and $0<y\le 10$.

Find the equations of the lines on which will lie the points where the solution curves to the differential equation have (i) horizontal and (ii) vertical tangents.

On the graph above sketch:

i) the lines identified in part (a)

ii) the solution curve that passes through the point (8, 6)

iii) the solution curve that passes through the point (4, 6)

Consider the differential equation

 $\frac{dy}{dx}={\left(\frac{1}{e^{\sqrt{x}}\cos x}\right)}^2-\frac{y}{\sqrt{x}}$ 

with the boundary condition $y\left(\frac{\mathrm{\pi }}{3}\right)=0$. 

Apply Euler’s method with a step size of $h=0.01$ to approximate the solution to the differential equation at $x=\frac{20\mathrm{\pi }+3}{60}$.

It can be shown that the exact solution to the differential equation with the given boundary condition is

$y=\frac{\tan x-\sqrt{3}}{e^{\sqrt[2]{x}}}$ 

(i) Compare your approximation from part (a) to the exact value of the solution at $x=\frac{20\mathrm{\pi }+3}{60}$.

(ii) Explain how the accuracy of the approximation in part (a) could be improved.

A particle moves in a straight line, such that its displacement $x$ at time $t$ is described by the differential equation

$x=\frac{t\sin \left(t^2\right)}{\cos x},t\ge 0$ 

At time $t=0,x=-\frac{\mathrm{\pi }}{3}$ 

By using Euler’s method with a step length of 0.2, find an approximate value for $x$ at time $t=0.6$.

Solve the differential equation with the given boundary condition.

Hence find the percentage error in your approximation for $x$ at time  $t=0.6.$

Consider the first-order differential equation

 $\frac{dy}{dx}+\frac{1}{2x}=\sin 3x\cos 3x$ 

By first finding the general solution to the equation, solve the equation for the case that  y=0   when $x=\frac{\mathrm{\pi }}{2}$.

Use separation of variables to find the general solution of each of the following differential equations:

$\frac{dy}{dx}=\frac{3y^4}{4x^3}$

$\frac{dy}{dx}=\frac{x^2}{y\left(\mathrm{\pi }-x^3\right)}{e}^{y^2}$

Solve each of the following differential equations for $y$ which satisfies the given boundary condition.

$\cos \mathrm{\pi }{x}^4\frac{dy}{dx}={\left(\frac{x}{y}\right)}^3\tan \mathrm{\pi }{x}^4;y\left(0\right)=-3$

$\left(\frac{e^{x^2}}{\cos y}\right)\frac{dy}{dx}=x^2\cos y;y\left(0\right)=\frac{3\mathrm{\pi }}{4}$

The evil Galactic Imperium has been spreading through the galaxy, taking over larger and larger volumes of galactic space as time goes on.  The area of space controlled by the Imperium at any point in time may be modelled as a sphere centred on the capital planet Merekhty.

Representatives of the Star Rebellion are on the planet Nezal, attempting to convince the planet’s inhabitants to join the rebellion.  Nezal lies 16.2 kiloparsecs (kpc) away from Merekhty, however, and because of that great distance the inhabitants of the planet believe it will be a very long time before they need to worry about the Imperium’s expansion.

As the Rebellion’s Chief Mathematician, you have been given the job of preparing a report on the expansion of the Imperium in relation to Nezal.  Based on your research, you believe that at any time, , the rate of expansion of the volume of space controlled by the Imperium, , is inversely proportional to the square of the cube root of the volume of space already controlled by the Imperium at that time.

Given that one year ago the Imperium controlled 8 cubic kiloparsecs of galactic space, whereas now it controls 2197 cubic kiloparsecs, determine how many more years it will be before Nezal falls within the Imperium’s sphere of control.

As the atoms in a sample of radioactive material undergo radioactive decay, the rate of change of the number of radioactive atoms remaining in the sample at any time $t$ is proportional to the number, $N$, of radioactive atoms currently remaining.  The amount of time, $\lambda$, that it takes for half the radioactive atoms in a sample of radioactive material to decay is known as the ­half-life of the material. 

Let $N_0$ be the number of radioactive atoms originally present in a sample. 

By first writing and solving an appropriate differential equation, show that the number of radioactive atoms remaining in the sample at any time $t\ge 0$ may be expressed as

$N\left(t\right)=N_0{e}^{-\frac{\ln 2}{\lambda }t}$

Plutonium-239, a by-product of uranium fission reactors, has a half-life of 24000 years.

For a particular sample of Plutonium-239, determine how long it will take until less than 1% of the original radioactive Plutonium-239 atoms in the sample remain.

The diagram below shows the slope field for the differential equation

$\frac{dy}{dx}=\cos \left(x-y\right)$ 

The graphs of the two solutions to the differential equation that pass through the points $\left(0,\frac{\mathrm{\pi }}{3}\right)$ and $\left(0,\mathrm{\pi }\right)$ are shown.

Explain the relationship that must exist between $x$ and $y$ for $\frac{dy}{dx}$=0 to be true.

For the two solutions given, the local minimum points lie on the straight line $L_1$and the local maximum points lie on the straight line $L_2$. 

Find the equations of  (i) $L_1$ and  (ii) $L_2$,  giving your answers in the form $y=mx+c$.

Consider the differential equation

 $\frac{dy}{dx}=\frac{5}{\sqrt{63+11x^2-2x^4}}-\frac{2xy}{2x^2+7}$ 

with the boundary condition $y\left(-\frac{3\sqrt{2}}{2}\right)=1$.

Apply Euler’s method with a step size of  $h=0.2$ to approximate the solution to the differential equation at $x=\frac{2-3\sqrt{3}}{2}$ .

It can be shown that exact solution to the differential equation with the given boundary condition is

$y=\frac{5{\sin }^{-1}\left(\frac{x}{3}\right)+4+\frac{5\mathrm{\pi }}{4}}{\sqrt{2x^2+7}}$

(i) Compare your approximation from part (a) to the exact value of the solution at $x=\frac{2-3\sqrt{2}}{2}$.

(ii) Explain how the accuracy of the approximation in part (a) could be improved.

A particle moves in a straight line, such that its displacement $x$ at time $t$ is described by the differential equation

 $x=\frac{t\left(t^2+1\right)e^{-3x}}{\left(2t^2+1\right)\left(2t^2+3\right)},t\ge 0$

At time $t=0$, $x=0$.

By using Euler’s method with a step length of 0.1, find an approximate value for $x$ at time $t=0.3$.

Solve the differential equation with the given boundary condition.

Hence find the percentage error in your approximation for $x$ at time $t=0.3$.