Lesson 92 — Separable ODEs

Solve $\dfrac{dy}{dx} = 5y$.

Solve the IVP $\dfrac{dy}{dx} = -\dfrac{y}{2}$, $y(0) = 4$.

Solve $\dfrac{dy}{dx} = xy$.

Solve $\dfrac{dy}{dx} = \dfrac{x}{y}$, $y(0) = 2$.

Solve $\dfrac{dy}{dx} = e^{x-y}$.

Solve $\dfrac{dy}{dx} = (1 + y^2)\cos x$.

Solve $\dfrac{dy}{dx} = \dfrac{x^2}{y}$, $y(1) = 2$.

Solve $y' = y\sqrt{x}$.

Solve $\dfrac{dy}{dx} = \dfrac{\cos x}{y}$.

Solve $\dfrac{dy}{dx} = \dfrac{e^{2x}}{y}$.

Solve $\dfrac{dy}{dx} = -2xy^2$.

Solve $\dfrac{dy}{dx} = y^2 - 1$ via partial fractions.

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

Verify that $y = \dfrac{1}{1+e^{-x}}$ solves $y' = y(1-y)$.

Solve $\dfrac{dy}{dx} = \dfrac{2x}{1+y^2}$.

Solve $y'\sin x = y\cos x$.

Solve $\dfrac{dy}{dx} = y\tan x$, $y(0) = 1$.

Solve $y'\,e^x = y$.

Solve $\dfrac{dy}{dx} = \dfrac{y}{x^2}$, $y(1) = e$.

Solve $y' = \sqrt{1 - y^2}$. Discuss the domain and singular solutions.

Radioactive decay: ${}^{14}$C has a half-life of 5730 years. What percentage remains after 10,000 years?

RC capacitor discharge: $V(0) = 12$ V, $R = 1\,\text{k}\Omega$, $C = 100\,\mu\text{F}$. Find $V(t)$.

A 100 L tank with pure water receives 5 L/min of brine at 10 g/L and drains 5 L/min. What is the concentration after 30 min?

Bacterial colony doubles every 3 h. How long to grow 100 times?

Newton's cooling: coffee at 90°C in a 20°C room reaches 70°C in 5 min. When does it reach 30°C?

Fall with linear resistance: $\dot v = g - kv$, $v(0) = 0$. Solve and determine terminal velocity $v_\infty$.

Drug concentration: $\dot C = -0.1C$, $C(0) = C_0$. How long until it drops to 50% of the initial dose?

Investment with continuous interest at 5% p.a.: $\dot S = 0.05 S$. How long to double the capital?

Show that $y \equiv 0$ is a solution to $y' = y^2$. Does it belong to the general family? Justify.

For $y' = y^{2/3}$, $y(0) = 0$, show that there are infinite solutions. Why does Picard's uniqueness fail?

Why does $\displaystyle\int \dfrac{dy}{y} = \ln|y| + C$ use the absolute value?

For $\dot y = y(1-y)$, identify the equilibria and classify them as stable or unstable.

Which of the forms below for $dy/dx = F(x,y)$ corresponds to a separable ODE?

Solve $y' = (x+y)^2$ via substitution $u = x+y$.

Show that $\dot y = y^2$, $y(0) = y_0 > 0$ blows up at finite time $T = 1/y_0$. Confirm with Osgood's criterion.

Bernoulli equation $y' + P(x)y = Q(x)y^n$. Show that the substitution $u = y^{1-n}$ transforms it into a linear ODE.

Outline the proof of the Picard-Lindelöf theorem for $y' = f(x,y)$, $y(x_0) = y_0$, with $f$ continuous and Lipschitz in $y$, via Picard iteration.

For $\dot y = h(y)$ with $h(y) > 0$ for all $y \geq y_0$, show that the solution is global if and only if $\displaystyle\int_{y_0}^{+\infty} \dfrac{dy}{h(y)} = +\infty$ (Osgood's criterion).

Solve $\dfrac{dy}{dx} = y^2 e^x$.

Solve $\dfrac{dy}{dx} = y^2 - 4$ via partial fractions. Identify singular solutions.

Solve $\dfrac{dy}{dx} = \dfrac{x^2}{1+y^2}$.

Solve $\dfrac{dy}{dx} = e^{-y}\sin x$.

Logistic growth: $\dot P = 0.06\,P(1 - P/500)$, $P(0) = 50$. When does the population reach half the carrying capacity?

Qualitatively analyze $\dot y = y(2-y)$ without solving explicitly: identify equilibria, stability, and behavior of solutions for different initial conditions.

For $\dot y = y^p$ with $y(0) = y_0 > 0$, determine for which values of $p$ finite-time blow-up occurs. Calculate $T$ in that case.