Ordinary Differential Equations: 30+ Hours!

Rewrite separable differential equations as f(y) dy = g(x) dx, integrate to y^2 = x^2 + K, then solve for y with ± and verify the solution.

Solve a differential equation by separation of variables, cross-multiplying to get y dy = x^2 dx, then integrate, rename constant as k, and take square root to yield two solutions.

Use separation of variables to solve the differential equation with respect to t, integrating both sides and applying arctangent, yielding y(t) = 2 tan(2 t^2 + C).

solve a differential equation by separation of variables, arranging into dx and dt terms, using completing the square and arctan to isolate x as a function of t.

Derive the exponential growth or decay model by setting dy/dt = k y and separating variables, then apply the initial condition to obtain y(t) = y0 e^{k t}.

This lecture demonstrates solving a differential equation by using a homogeneous equation and the substitution y = v x with v = y/x. It yields a separable equation.

Solves a homogeneous differential equation by substituting y = v x, separating variables and integrating, revealing the regular solution and the singular solution y = 0.

Solve a homogeneous first-order differential equation by substituting v = y/x, reducing to a separable form, integrating, and back-substituting to obtain the solution.

Transform a differential equation to a separable form, then solve by substitution and integration, noting the special case z = -1/2 yields a distinct solution.

Solve a differential equation by reducing to a homogeneous form, using y = v x, solving a linear system to remove constants, and addressing singular solutions and partial fraction integrals.

Explain exact differential equations by finding a potential function f(x,y) with partials f_x = M and f_y = N, then set f(x,y) = C as the implicit solution.

Verify the exactness of a differential equation in t and x and solve by constructing a function f(t, x) with g(x) determined from the x-derivative.

Identify that the differential equation is not exact, apply an integrating factor to make it exact, and solve for the general solution f(x,y) = C.

Identify that the differential equation is not exact. Apply an integrating factor x to make it exact and solve via a potential function f(x,y)=C for the general solution.

The lecture teaches checking exactness, testing integrating factors from a table, applies μ = 1/(x y^2) to make the equation exact, and derives the general solution F(x,y)=C.

Use an integrating factor to turn the differential equation into an exact one, yielding F(x,y) = -x^3/y + 2y^3/3; applying y(1) = -1 gives the constant C = 1/3.

Multiply the non-exact differential equation by an integrating factor mu(x+y) to make it exact, derive mu(t) with t = x+y using the chain rule, and verify exactness.

Explore a generalized method to find an integrating factor mu(x,y) for a given ordinary differential equation, showing the factor depends only on x y and yields an exact equation.

Learn the linear differential equation y' + a(x) y = b(x) using the integrating factor method; take B(x) as integral of b(x), and obtain y = 2 + C e^{−x^2}.

Solve a linear differential equation with respect to t, using integration by parts and an initial condition y(0)=1. Describe how the constant is determined and the solution form.

Understand Bernoulli's equation in the form y' + p(x) y + q(x) y^n. Use the substitution v = y^{1-n} to obtain a linear equation in v, then recover y.

Solve an initial value problem for a Bernoulli equation by substituting v = 1/y to linearize it, solve for v(x), then revert to y and apply the initial condition y(1)=5/2.

Choose a particular y1 for a Riccati equation and set y = y1 + 1/z; solve the linear equation for z to obtain y's general solution, extra y = y1.

Use Riccati techniques: find a particular solution y1 and set y = y1 + 1/z; with y1 = x, z' = -1 gives y = x - 1/(x - C).

Solve a Riccati equation by guessing a particular solution y1, set y = y1 + 1/z to obtain a linear equation for z, then recover the general solution for y.