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.

Apply separation of variables to solve the differential equation with y(2)=1; determine C=0, obtain the solution, and note domain restrictions x ≠ 1 and y ≠ 0 (y=0 fails).

Solve differential equation by separation of variables, factoring into y-3 and x+3, integrate 1/(y-3) and x^2/2+3x, use y(1)=-1 to determine C, and note the restriction y not equal to 3.

Rearrange and factor the equation to separate variables, then integrate under nonzero assumptions; identify a singular solution y = -2 alongside the normal solution.

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.

Solve a differential equation with initial condition y(pi)=1 using separation of variables, set c=0, and obtain y = -1/cos x while noting y not equal to zero.

Solve a differential equation by separation of variables with initial condition y(0)=1, perform integration, note y=0 is a singular solution ruled out, and derive the solution without isolating y.

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}.

Explore homogeneous ordinary differential equations, defined when M(x,y) and N(x,y) are homogeneous of the same degree, and solve via substitution y = v x, then separate and back-substitute.

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.

Shows equation is homogeneous and solved by y = v x to yield a separable form, using partial fractions to obtain the solutions y = x and y = -3x.

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 differential equation by substituting y = v x, verify M and N are homogeneous of order two, and derive the main solution plus two singular solutions.

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.

Solve a homogeneous differential equation using the substitution y equals v x or x as a function of y, yielding a separable form and singular solution y equals 0.

Classify the differential equation as homogeneous, then solve by letting y = v x, separating variables, integrating, and replacing v with y/x to obtain the solution.

Apply substitution to linear-in-x,y differential equations to form homogeneous. Case 1 is separable, case 2 yields a homogeneous equation via a linear system and back-substitution to recover x and y.

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 linear differential equation in x and y by using a z substitution, reduce to a separable form, integrate, and back-substitute to express the solution with a constant.

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.

Learn to transform and solve first-order differential equations by cross-multiplication, converting to homogeneous or separable forms, using substitution y = v x, and identifying singular solutions.

Transform a linear differential equation into a homogeneous equation of degree 1, apply the substitution v = y/x, separate variables, and back-substitute to obtain the general solution.

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.

Check ∂M/∂y = ∂N/∂x for exactness, integrate M with respect to x, determine g(y) from the y-derivative, and express the implicit solution F(x,y)=C.

Demonstrates solving an exact differential equation by verifying exactness, integrating M with respect to x, determining g(y), and obtaining F(x,y) = constant as the solution.

Check the exactness of a differential equation by verifying that ∂M/∂y equals ∂N/∂x, then solve using the potential function f with xy = C.

Check exactness by equating the derivatives with respect to y and x, then construct the potential function by integrating M with respect to x and identifying the y terms.

Verify an exact differential equation by checking M_y = N_x, integrate with respect to x, and obtain the general solution f(x,y)=c through partial derivatives.

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.

determine k for exactness, find k=1, then build a potential function f with fx and fy, yielding f(x,y)=x^3+(e^{xy})/y+(y^4)/2 and f=C.

The lecture introduces the integrating factor to turn a non-exact differential equation into an exact form, showing when it depends on x or y and how to compute it.

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.

Demonstrate that a differential equation is not exact, apply the integrating factor y^x, and derive an exact form to obtain the general solution.

Explore converting a non-exact differential equation to an exact one with an integrating factor, verify exactness, and solve for f(x,y)=C by integrating with respect to y.

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.

Analyze solving a differential equation by testing exactness, applying integrating factors such as e^{-2x} or (x^2+y^2)^{-1}, and deriving the potential function F(x,y) to obtain an exact form.

Turn a non-exact differential equation into an exact one by finding an integrating factor, then derive the solution using M and N with respect to x and y.

Check whether the differential equation is exact, find an integrating factor as a function of y, multiply through to obtain an exact equation, and derive the general solution f(x,y)=C.

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.

Derive an integrating factor μ(xy) for a non exact ode by applying chain and product rules, and show μ(xy) = (xy)^2 renders the equation exact.

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.

Use an integrating factor of the form x^α y^β to solve a non-exact ODE, derive mu(T) with T = x y^3, and find alpha = -1, beta = -3.

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.

Find an integrating factor mu depending on x+y to make the differential form exact, derive the exactness condition ∂(mu M)/∂y = ∂(mu N)/∂x, and express mu as exp(G(x+y)).

Derive an integrating factor mu of x/y for a general first-order differential equation by enforcing cross-derivative equality and applying the product rule.

Explore solving linear first-order differential equations using integrating factors, converting to y' + a(x) y = b(x) and applying the closed formula y = e^{-A(x)} [ ∫ e^{A(x)} b(x) dx + C ].

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}.

Learn to solve a linear differential equation using an integrating factor, transforming y' - (1/x) y = x^2 + 3x - 2 for x>0.

Convert the differential equation to the linear form, identify A(x) and B(x), compute A(x) as the integral of x and B(x) from the expressions, and substitute back to solve.

Transform and solve a differential equation by converting to linear form, dividing by x^3 to isolate y', evaluating integrals for asterisk and double asterisk, and simplifying with exponent rules.

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.

solve a linear differential equation with integrating factor, using A(x)=cot x and B(x) with cos x terms, and apply ∫ cot x dx = ln sin x to derive solution.

Solve a linear first-order differential equation using an integrating factor: y' + cot x · y = 1; compute integrals of cot x and csc^2 x to obtain the solution.

Rewrite the equation in standard linear form, compute a(x) and B(x), and apply z(pi)=0 to obtain z(x)=sin x / x^2.

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 a Bernoulli differential equation by converting it to a linear form. Apply the substitution v = y^{-2}, solve, then back-substitute to obtain y(x).

solve the original differential equation, a bernoulli type, by substituting v = y^(1-n) to obtain a linear equation in v, then back-substitute to get y.

This lecture solves a Bernoulli differential equation by substituting v = sqrt(y), converting to a linear equation, applying the integrating factor, and recovering y = v^2 with x positive.

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.

Substitute v = z^-2 to turn the differential equation into a linear Bernoulli form, using p(x) = -cot x and q(x) = 1/ sin x (0 to pi).

Explore Riccati equations in ordinary differential equations by guessing a particular solution, reducing to a linear equation for z, and obtaining the general solution y = y1 + 1/z.

Solving a Riccati equation by guessing a particular solution y1(x) and using y = y1 + 1/z to obtain z, yields y = y1 + 1/z.

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.

This lecture explains the existence and uniqueness theorem for first order ODEs and initial value problems, showing when solutions exist, when they are unique, and examples with various solution counts.

Construct two solutions for a given initial value problem and verify initial condition and differential equation to explain why this does not violate the uniqueness theorem, due to f_y's discontinuity.

Examine an initial value problem, apply separation of variables to solve a differential equation, verify existence but not uniqueness conditions, and establish a unique solution with constant C.

Apply the existence and uniqueness theorem to an initial value problem, showing the zero solution y(x)=0 is unique when f and f_y are continuous in a rectangle around (4,0).

Apply the existence and uniqueness theorem to the initial value problem for a polynomial differential equation, proving any solution is bounded below and is increasing with positive derivative.