A Complete Guide to 1st-Order Ordinary Differential Equation

Define differential equations and classify them by type, order, and linearity, distinguishing ordinary differential equations from partial differential equations, with examples.

Verify if a function solves a differential equation by computing derivatives and substituting them into the equation; shown with a first-order linear differential equation and a second-order linear differential equation.

Explore two special first-order differential equations and solve for y as function of x using separable variables. Separate variables, integrate, and determine the constant from initial conditions to obtain solution.

Study initial value problems for first-order and higher-order odes, using separable variables and integration with initial conditions to derive specific solutions, such as y = tan(4x - 3π/4).

Solve first-order differential equations where the derivative is proportional to the dependent variable using separation of variables, yielding exponential growth or decay.

Explore real-life applications of first-order ODE model I, including exponential growth in bacterial populations, radioactive decay with half-life, and carbon dating to estimate fossil age.

Solve first-order ODEs with an upper or lower bound using separation of variables. Find y(t)=n-(n-y0)e^{-kt} for the upper bound and y(t)=n+(y0-n)e^{-kt} for the lower bound, both approaching the bound.

Apply Newton's law of cooling to model a cake cooling from 300° to 70°, solving with k from T(3)=200 ≈ 0.19, and showing it approaches 70° around 30 minutes.

Explore the logistic equation, a nonlinear first-order ODE solved by separation and partial fractions, modeling growth toward the upper bound n with an inflection at y = n/2.

Explore the logistic model three: bounded growth; study a campus outbreak with upper bound 1000, x(4)=50, and predict x(6) around 276.

Explore five population models with first-order ODEs, including exponential growth and decay, logistic growth with upper bounds, and scenarios of steady state, extinction, or doomsday.

Solve general first-order linear odes with the integrating factor. Multiply by mu(t)=e^{∫p(t)dt}, turn the left side into a derivative, then integrate to get y; example yields y=t^3/5+C/t^2.

Solve first-order linear ODEs with integrating factors, illustrate with two numerical examples, and analyze initial-condition cases for compound interest and population models.

Explore a first-order linear differential equation modeling a tank mixing problem with inflow and outflow, yielding a solution for salt amount y(t), its maximum at t=60, and empty time t=200.

Explore first-order ODE applications in series circuits, deriving LCR, RC, and LR models via Kirchhoff's law and solving with integrating factors to obtain q(t) and i(t) under E(t).

Analyzing a projectile from Earth using a first-order differential equation derived from Newton's gravity, the lecture solves for velocity as a function of altitude, maximum height, and escape velocity.

Apply Newton's second law to a growing hailstone with time-varying mass, derive a first-order ODE, and show acceleration equals a constant g/4 independent of initial velocity.

Solve a homogeneous first-order ode for motion using u = y/x and separation of variables, and examine cases k = w/v0 with k<1, k=1, k>1 affecting arrival at the origin.

Torricelli's law models fluid exiting a hole and derives a separable differential equation linking hole area, contraction coefficient, and velocity v = sqrt(2 g y) with height y.

Use a first-order ordinary differential equation to model fluid motion in a hemispherical bowl, deriving area and height relations and solving for the time to empty and hole radius.

Model motion with a water clock by deriving the profile f(x) that, when revolved about the y-axis, creates the tank shape, and compute the hole radius to achieve 12-hour duration.

present four methods for solving first-order ODEs: special case f(x), separable, linear with integrating factor, and homogeneous via y = u x, with a preview of exact first-order ODEs.

Identify exact first-order ODEs by recognizing left-hand side as an exact differential df. Solve by finding f with ∂f/∂x = m and ∂f/∂y = n, then f = c.

Apply method five for first-order odes by testing exactness with dm/dy = dn/dx, then find a function f with f_x = m and f_y = n, giving f(x,y) = constant.

Use method six to turn a non-exact differential equation into an exact one by applying an integrating factor mu, which may depend on x or y, yielding a separable solution.

Apply method six to turn non exact differential equations into exact ones using integrating factors, and derive an implicit solution from two examples.

Solve Bernoulli's equation by transforming it into a linear ODE via the substitution y^{1−n}. Handle n=0 and n=1 with linear and separable forms, and reduce cases to a linear ODE.

This example uses method VII for a first-order ODE with y^2, converts to Bernoulli form, solves linear ODE to yield y = 1/(c x − x^2) for x ≠ 0.

This lecture presents method eight: solving first-order ODEs by substitution. It uses a substitution like u=3x+2y to obtain a separable equation and a relation between x and y.