Demonstrate the variable separation method for first-order, first-degree differential equations, separating the dependent and independent variables, then integrating both sides and applying the constant of integration.

Apply the variable separable method to a sphere whose volume decreases proportionally to surface area, radius 2 cm initially and 1 cm after 3 months, predicting zero at 6 months.

Apply the variable separable method to solve Newton's cooling differential equation with ambient temperature 25 degrees centigrade, finding the body temperature after 30 minutes—approximately 31.4 degrees centigrade.

Apply the variable separable method to solve Newton's law of cooling with ambient zero degrees centigrade and initial 21 degrees, finding the time to reach 5 degrees about 1.4 minutes.

Separate variables and integrate to solve the differential equation, yielding x^2/2 + c, with the constant undetermined due to missing initial conditions.

Apply separation of variables to a bacteria growth model with rate proportional to N; find k from doubling in two hours and compute the tripling time, approximately 3.16 hours.

Solve a differential equation by the variable separable method using example 6; introduce z = x + y + 1, separate, integrate, and apply y(0)=1 to obtain the solution.

Learn to solve a differential equation by variable separation, transforming it to a separable form and applying a substitution to integrate, yielding an implicit solution without initial conditions.

Demonstrates solving a separable differential equation via substitution to separate variables, using ∫ dx/(a^2+x^2) = (1/a) arctan(x/a), yielding Y = C + K arctan(X/K).

Identify homogeneous differential equations of the first degree. Solve by substituting y = v x to reduce to a separable equation, yielding a family of circles through the origin.

Solving a homogeneous differential equation of degree zero by substituting v = y/x, converting to a separable form, and integrating to obtain a solution in terms of ln(y/x).

Solve homogeneous differential equations by recognizing zero-degree homogeneous functions, applying Y = vX, and using separation of variables to derive the solution.

Explain the linear differential equation dy/dx + P(x) y = Q(x) and solve with integrating factor μ(x) = e^{∫P dx}, yielding y μ = ∫ μ Q dx + C.

Use the integrating factor method to solve a linear differential equation, multiply through, integrate, and apply the initial condition to determine the constant for the complete solution.

Convert the differential equation to standard form with x as a function of y, apply the integrating factor, and solve; then use the product-rule integration shortcut for multiplying functions.

Explain solving a linear differential equation by converting it to standard form, computing the integrating factor from p(x), and deriving the solution.

Convert a nonlinear differential equation into a linear form via substitution, apply the integrating factor e^{x^2}, and solve for the original variable y.

Transform the equation into a standard linear form, apply the integrating factor 1/x, and integrate to obtain the general solution with a term in x plus a 1/x term.

Master solving linear first-order differential equations with the integrating factor, converting to standard form when the right-hand side is a function of x, as shown in example 6.

Convert the equation to standard linear form, determine the integrating factor as log x, and solve for y using substitution, yielding y in terms of log x.

Learn how to solve a linear differential equation by converting it to standard form and applying an integrating factor. Use partial fractions to derive the general solution.

Master solving differential equations by inspection, rearranging terms to a single differential, and integrating to obtain solutions like log(xy) or xy, with constants of integration.

Demonstrates solving a non-linear differential equation by inspection, combining terms to form a single derivative, then dividing by x^2 and integrating to obtain the general solution.

this lecture demonstrates solving a nonlinear differential equation by inspection, avoiding linearization, and forming a single differential to obtain a first integral, yielding a constant relation involving y.

Learn to solve a differential equation by inspection, using substitution with x^2 + y^2 and the derivative of the inverse of y over x to derive the solution.

Explore Bernoulli's differential equation, a linear equation with a power term, and convert it into a standard linear form by substituting y = w^{1-n}, then solve using the integrating factor.

Convert the Bernoulli's differential equation in example 1 into a standard linear form using substitution. Apply the integrating factor to derive the solution, then substitute back to x and y.

Identify exact differential equations in the form M dx + N dy = 0 by ∂M/∂y = ∂N/∂x, then integrate to find F(x,y) = C.

Check exactness of M dx + N dy = 0 using M_y and N_x; then integrate M w.r.t x and N w.r.t y to obtain the solution with C.

Solve an exact differential equation (example 2) by using M dx + N dy = 0, verify exactness, and integrate to obtain the implicit solution.

solve an exact differential equation by verifying exactness, computing M and N, and integrating to find the potential function, while choosing the easier integration route.

This lecture shows how to verify an exact differential equation and derive the solution by integrating and applying the constant of integration, yielding x + x/y = C.

Convert a non-exact differential equation to an exact one by multiplying by an integrating factor such as 1/x^2 or 1/y^2 (or 1/(x^2+y^2)); then integrate to obtain the solution, noting non-uniqueness.

Apply rule 1 for equations M dx + N dy, using integrating factor 1/(M x + N y) when M x + N y ≠ 0 to make it exact.

Evaluate exactness of a cubic homogeneous differential equation, apply rule 1 to compute the integrating factor 1/(x^2 y^2), convert to an exact form, and integrate to obtain the solution.

Apply rule no. 2 to convert a non-exact differential equation into an exact form using an integrating factor, derive the potential function, and confirm the solution by checking exactness.

Explore rule no 3 for non-exact differential equations, compute an integrating factor, convert to an exact form, and solve by integrating with respect to x after multiplying through.

Rule four shows how to convert a non-exact differential equation into an exact one using an integrating factor, illustrated by a worked example.

Identify Clairaut's differential equation as yielding a linear function of x: y = C x + f(C); differentiate to show x is constant and determine C from initial conditions.

Explore higher order linear differential equations, identify homogeneous and nonhomogeneous cases, and learn that the general solution equals the complementary function plus a particular integral.

Learn to convert a differential equation to an algebraic auxiliary equation, determine roots, and build the complementary function for real, complex, and repeated roots using exponential, trigonometric, and hyperbolic forms.

Solve a third-order homogeneous differential equation by converting to a differential-operator form, factoring the cubic to identify roots, and formulating the complementary function with repeated roots.

Solves a homogeneous differential equation by converting to a differential operator form, solving the resulting algebraic equation for roots (including complex conjugates), and deriving the complementary function.

Convert a homogeneous differential equation to a differential-operator form, factor it as (D+1)^2 (D-2)^2, and derive the complementary function, showing repeated roots at -1 and 2.

Learn to solve linear higher-order differential equations with constant coefficients by deriving the complementary function from the auxiliary equation and obtaining the particular integral via the operator method for exponentials.

Analyze a type one differential equation, derive its auxiliary equation, and form the complementary function with a particular integral; address the zero-denominator special case by multiplying by x and differentiating.

Solve y''+9y=x by finding r^2+9=0, yielding y_c=c1 cos 3x + c2 sin 3x; convert the forcing to exponential form to find a particular integral and combine for the solution.

Determine the complementary function and particular integral for a second-order differential equation with constant coefficients, giving the solution y = C1 cos 2x + C2 sin 2x + 4.

Replace D^2 by -a^2 for sinusoidal forcing to obtain the particular integral; in y''+9y=sin 2x, the PI is sin 2x/5, then add the complementary function.

solve a resonance case of a second-order differential equation with cos 2x forcing by deriving the complementary function and a (x/4) sin 2x particular integral.

Identify the complementary function from the auxiliary equation roots, then compute a sinusoidal particular integral for the right-hand side, yielding y = y_c + (cos x)/10 - (sin x)/5.

Learn to compute the particular integral for algebraic functions using binomial expansions of operator expressions, and apply it alongside the complementary function to solve a differential equation.

Determine the complementary function from (D-2)^2 for a repeated root, using C1 and C2 x, and derive the particular integral for the AL-JABRI function with binomial expansion.

We compute the particular integral for exponential functions multiplied by cosine and sine, using operator substitution (D plus alpha) and the auxiliary equation to obtain the complete solution.

Identify the differential equation type by finding repeated roots to form the complementary function, then compute the particular integral using the operator method, and combine them for the solution.

Explore particular integrals of type 5 for equations with a sinusoidal multiplier, deriving the complementary function from d^2+9 and computing a specific particular integral for sinusoids.

Solve a second-order differential equation with cos 2x forcing by deriving the complementary function from r^2+4=0 and applying a resonance-aware particular integral, then combine for the full solution.