Lesson 91 — Introduction to Ordinary Differential Equations

Classify $y''' + y'' - y = 0$: determine the order, if it is linear, and if it is homogeneous.

Classify $y' + xy = 0$: order, linear/non-linear, homogeneous.

Classify $y' + y^2 = x$: order and linear/non-linear.

Classify $y'' + y = \sin x$: order, linear/non-linear, homogeneous.

Verify that $y = e^{2x}$ is a solution to $y' = 2y$.

Verify that $y = \sin x$ is a solution to $y'' + y = 0$.

Verify that $y = x^2 + 3$ is a solution to the IVP $y' = 2x$, $y(0) = 3$.

Verify that $y = e^{-t}\cos t$ is a solution to $y'' + 2y' + 2y = 0$.

Show that $y(t) = Ce^{-kt}$ is the general solution to $y' = -ky$.

Free fall is modeled by $y'' = -g$ (constant gravitational acceleration). Find the general solution by double integration.

What is the general solution to $y'' - y = 0$?

What does an initial condition determine in the solution of an ODE?

Solve $y' = 3x^2$, $y(0) = 5$.

Solve $y' = \sin x$, $y(0) = 1$.

Solve $y'' = 6x$, $y(0) = 1$, $y'(0) = 2$.

Solve $y' = ky$, $y(0) = y_0$. Express in terms of $k$ and $y_0$.

Solve $y' = 2y$, $y(0) = 5$. Calculate $y(3)$.

Solve $y' = -0.1\,y$, $y(0) = 100$. Calculate $y(20)$.

Solve $y' = e^x$, $y(0) = 0$.

Solve $y'' = 0$, $y(0) = 1$, $y'(0) = -3$.

Solve $y' = 1/x$, $y(1) = 0$ (domain $x > 0$).

Solve $y'' = -y$, $y(0) = 1$, $y'(0) = 0$.

Solve $y'' + 2y' + 2y = 0$, $y(0) = 0$, $y'(0) = 1$.

Capacitor discharges: $V' = -V/(RC)$. For $V_0 = 12$ V and $RC = 1$ s, calculate $V(2)$.

Bacterial colony doubles every hour. Initial population: 100. Write the ODE and calculate $N(5)$.

Investment of R$ 1,000 at 5% p.a. with continuous interest. Write the ODE and calculate the amount in 10 years.

Coffee at 90 °C, room 25 °C, $k = 0.04$ min$^{-1}$. Write the ODE, solve, and determine how long it takes for the temperature to reach 50 °C.

Carbon-14 ($\tau_{1/2} = 5730$ years). A fossil has 25% of the initial C-14. Calculate its age.

Medication: half-life of 6 h, dose 200 mg. Write the ODE and calculate how much remains after 18 h.

Financial investment yields Selic of 14.75% p.a. with continuous compounding. In how many years does the capital double?

Simplified epidemic: $I' = rI(1 - I/N)$ (logistic equation). Identify the equilibria and describe the solution behavior.

Fall with air resistance: $m\dot{v} = mg - kv$ ($k > 0$). Calculate terminal velocity $v_\infty$ (when acceleration ceases).

Iodine-131 ($\tau_{1/2} = 8$ days). Write the ODE and calculate how much remains of 100 g after 24 days.

An asset depreciates 3% per year continuously. Write the ODE and express the value after 5 years in terms of the initial value $P_0$.

(Forensic) Body found at 10 PM with temperature 32 °C. Room at 21 °C, $k = 0.374$ h$^{-1}$. Normal body temperature: 37 °C. Write the ODE and find $k$ using the given conditions.

What does the Picard-Lindelöf Theorem guarantee about the ODE $y' = f(x, y)$, $y(x_0) = y_0$?

Why does the general solution of an $n$-th order ODE have exactly $n$ arbitrary constants? How does this relate to the number of initial conditions required?

Explain why $y' = \sqrt{\lvert y \rvert}$, $y(0) = 0$ has infinitely many solutions. Which Picard-Lindelöf hypothesis is violated?

Prove that $y = Ce^{kx}$ is the only family of solutions to $y' = ky$ (up to choice of $C$). Hint: consider $z(x) = y(x)\,e^{-kx}$.

Prove that the solution to $y' = ky$, $y(0) = y_0$ is $y(t) = y_0 e^{kt}$, using the technique of separation of variables (preview of Lesson 92).