Lesson 93 — 1st-order linear ODEs

Solve $y' + y = 0$.

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

Solve $y' + 3y = e^{-x}$.

Solve $y' - y = e^{2x}$, $y(0) = 0$.

Solve $y' + 2xy = x$.

Solve $y' + \frac{1}{x}\,y = 1$, $y(1) = 0$.

Solve $x y' + 2y = x^3$.

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

Solve $y' + y\cot x = 2\cos x$.

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

Solve $y' + 4y = 8$ and indicate the steady state.

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

Solve $y' - 3y = 6e^{3x}$.

Solve $y' + \frac{2}{x}\,y = x^2$.

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

Solve $(1+x^2)y' + 2xy = 4x$, $y(0) = 0$.

Does $y = e^{-x}$ solve $y' + y = 0$?

Solve $y' = 2y + 4$.

Find $\mu(x)$ for $y' + (\sin x)y = \cos x$ and discuss if the solution has a closed form.

Solve $y' + 2y = t^2$ (independent variable $t$).

RC circuit: $R = 2\,\text{k}\Omega$, $C = 50\,\mu\text{F}$, $V_s = 12\,\text{V}$ (step at $t = 0$), $V_C(0) = 0$. Find $V_C(t)$ and $\tau$.

RL circuit: $L = 0.5\,\text{H}$, $R = 10\,\Omega$, $V = 5\,\text{V}$, $i(0) = 0$. Find $i(t)$ and $\tau = L/R$.

CSTR reactor: flow $Q = 2\,\text{L/min}$, volume $V = 100\,\text{L}$, input concentration $c_{in} = 5\,\text{g/L}$, $c(0) = 0$. Find $c(t)$.

Thermometer at $20\,\text{°C}$ immersed in water at $80\,\text{°C}$. Time constant $\tau = 30\,\text{s}$. When does it read $70\,\text{°C}$?

Bank account: zero initial balance, earns $r = 5\%$ per year (continuous) and receives a continuous deposit of 12,000/year. Calculate the balance after 10 years.

Heater: $\dot T = -k(T - T_{\text{amb}}) + P/C$, with $k = 0.1\,\text{min}^{-1}$, $T_{\text{amb}} = 20\,\text{°C}$, $P/C = 5\,\text{°C/min}$, $T(0) = 20\,\text{°C}$. Calculate $T_\infty$ and the time to reach 90% of $T_\infty$.

RC low-pass filter with input $u(t) = \sin(\omega t)$. Calculate the steady-state response amplitude for $\omega\tau = 0.1$, $\omega\tau = 1$, and $\omega\tau = 10$.

200 L tank contains 50 g of salt. Brine enters at 2 g/L at 4 L/min; mixture exits at 4 L/min. Amount of salt after 1 hour.

Show that if $p$ and $q$ are continuous on $I$, the IVP $y' + p\,y = q$, $y(x_0) = y_0$ has a unique solution.

The integrating factor is unique up to a multiplicative constant — does this affect the final solution?

State the superposition principle for $\mathcal{L}y = y' + p\,y$.

Bernoulli equation: $y' + y = y^2$. Use $u = 1/y$ to linearize and solve.

Solve $y' + y\tan x = \sec x$ via variation of parameters and confirm that the result matches the integrating factor method.

Riccati equation: $y' = y^2 - 2xy + x^2 - 1$. Verify that $y_1 = x + 1$ is a particular solution and use the substitution $y = x + 1 + 1/v$ to reduce it to a linear equation in $v$.

Rigorously demonstrate that $\mu(x) = e^{\int p(x)\,dx}$ transforms $y' + py = q$ into $(\mu y)' = \mu q$ and deduce the general solution formula.

Demonstrate that $y(x) = \int_{x_0}^{x} G(x,s)\,q(s)\,ds$, with $G(x,s) = e^{-\int_s^x p(t)\,dt}$, solves $y' + py = q$, $y(x_0) = 0$.