5.18 First-order differential equations questions for IB Maths AA

Write down the value of ${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}}$ when $x=0$.

Apply Euler's method, with $h=0.5$, to estimate the value of $y$ at $x=1$.

By solving the differential equation, show that $T=22+68e^{-kt}$.

After 5 minutes, the temperature of the soup is ${70}^{\circ }$C. Find the value of $k$.

Find the time it takes for the soup to cool to ${40}^{\circ }$C.

Let $z=y^{1-n}$. Show that the Bernoulli equation can be written as

Consider the differential equation ${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}}-{\displaystyle \frac{y}{x}}=-y^2$, for $x>0$, where $y={\displaystyle \frac{1}{2}}$ when $x=1$.

Use part (a) to show that $z={\displaystyle \frac{1}{y}}$ satisfies ${\displaystyle \frac{\mathrm{d}z}{\mathrm{d}x}}+{\displaystyle \frac{z}{x}}=1$.

Hence show that $y={\displaystyle \frac{2x}{x^2+3}}$.

Find the exact coordinates of the maximum point on the graph of $y={\displaystyle \frac{2x}{x^2+3}}$.

The logistic equation ${\displaystyle \frac{\mathrm{d}N}{\mathrm{d}t}}=rN(1-{\displaystyle \frac{N}{K}})$, where $r$ and $K$ are positive constants, models a population $N$ at time $t$. It is given that $N=N_0$ when $t=0$, where $0<N_0<K$.

Show that the logistic equation is a Bernoulli equation, stating the values of $n$, $P$ and $Q$.

Use the substitution $z={\displaystyle \frac{1}{N}}$ to show that $N={\displaystyle \frac{K{N}_0}{N_0+(K-N_0)e^{-rt}}}$.

Write down ${\displaystyle \underset{t\to \mathrm{\infty }}{\lim }N}$.

Consider the differential equation ${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}}+y=e^x\sqrt{y}$, where $y=1$ when $x=0$.

Show that the substitution $z=\sqrt{y}$ gives ${\displaystyle \frac{\mathrm{d}z}{\mathrm{d}x}}+{\displaystyle \frac{1}{2}}z={\displaystyle \frac{1}{2}}{e}^x$.

Hence find $y$ in terms of $x$.

Verify that $y=0$ also satisfies the differential equation in part (d), and suggest why the method of part (a) does not produce this solution.

Apply Euler's method, with a step length of $0.2$, to estimate the value of $y$ at $x=2$.

Show that ${\displaystyle \frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}}={\displaystyle \frac{1+\sqrt{x+y}}{2\sqrt{x+y}}}$, and hence determine whether your answer to part (a) is an underestimate or an overestimate of the true value of $y$.

Solve the differential equation.

Apply Euler's method, with a step length of $0.25$, to estimate the value of $y$ at $x=1$.

Show that, with a step length of ${\displaystyle \frac{1}{n}}$, where $n\in {\mathbb{Z}}^{+}$, Euler's method gives the estimate ${(1+{\displaystyle \frac{1}{n}})}^n$ for the value of $y$ at $x=1$.

By considering the concavity of the solution curve, explain why every estimate in part (c) is less than $e$.

By considering $\ln {(1+{\displaystyle \frac{1}{n}})}^n$, use l'Hôpital's rule to show that ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{1}{n})}^n=e}$.

Let $E(n)=e-{(1+{\displaystyle \frac{1}{n}})}^n$ be the error in the estimate.

Use your GDC to find $E(n)$ and $nE(n)$ for $n=10,20,40$ and $80$. Give your answers correct to four significant figures.

Hence describe the effect on the error of halving the step length, and suggest the limit of $nE(n)$ as $n\to \mathrm{\infty }$.

Use the Maclaurin series for $\ln (1+x)$ to show that $n\ln (1+{\displaystyle \frac{1}{n}})=1-{\displaystyle \frac{1}{2n}}+{\displaystyle \frac{1}{3n^2}}-\dots$, and hence show that $E(n)\approx {\displaystyle \frac{e}{2n}}$ for large $n$.

Use the approximation $E(n)\approx {\displaystyle \frac{e}{2n}}$ to estimate the least value of $n$ for which the error is less than $0.001$.

Use your GDC to find the least value of $n$ for which $E(n)<0.001$.