Differential Equations in an easy way!

Learn essential derivatives formulae and differentiation rules, including the chain rule, with examples on constants, powers, roots, logs, exponentials, trig and inverse functions, and upcoming integration.

Master the rules of differentiation, including constants, and the sum, difference, product, and the quotient rules. Explore the chain rule with examples like e^x, log, and sin 4x.

Explore the basics of integration and its link to derivatives. Master key rules and techniques such as substitution, integration by parts, and partial fractions.

Master the basic rules of integration: constants outside, linearity for sums and differences, and linear substitutions replacing x with a x + b. See examples like x^5 and sin(3x).

Master integration by parts and substitution with solved examples, covering indefinite and definite integrals, generalized rules, and common pitfalls in exponential and trigonometric integrals.

Explore definite integrals and their properties, including lower and upper bounds, splitting, bounds interchange with a minus sign, and change of variables to simplify calculations.

Apply definite integration properties to solve problems using even/odd symmetry and the a plus b minus x substitution, simplifying sine squared and cosine squared sums over symmetric bounds.

Learn to form and solve differential equations with the variable separable method, separating variables and integrating to yield solutions such as y = e^(x^2/2 + C).

The lecture teaches solving variable separable differential equations by integrating both sides, using initial conditions to obtain a complete solution, and deriving arctan forms from 1/(1+x^2) integrals.

Turn a non-separable differential equation into a separable form with substitution; differentiate, separate variables, and integrate, using u = x + y and u = 4x + y + 1.

Identify homogeneous first-order differential equations by equal total degrees of non-derivative terms, ignore derivatives, and solve using the substitution y = v x to obtain a separable form.

Identify homogeneous differential equation by degree two, substitute y = v x to obtain a separable form, arctan(v) = ln x + C and y/x = tan(ln x + C).

Determine a degree two homogeneous differential equation, apply the substitution y = v x, convert to a separable form and integrate to obtain log y = -1/x + c.

Explains solving a homogeneous differential equation using the substitution y = v x, converting it to a separable equation in v and x, then integrating and back-substituting v = y/x.

Understand exact differential equations m dx + n dy = 0, with ∂M/∂y = ∂N/∂x, and solve via ∫ M dx + ∫ N dy = C.

Explore exact differential equations and two equivalent solution methods, verify exactness using M dx plus N dy, and solve reducible cases with convenient integration strategies.

Learn how non-exact differential equations become exact using integrating factors, especially when M and N are homogeneous, via rule one: the integrating factor 1/(M x + N y).

Apply rule two to non-exact equations matching an x–y pattern, using integrating factor 1/(m x − n y) to become exact; rule three yields integrating factors from dM/dy − dN/dx.

Learn to reduce to exact differential equations by applying rule one, check exactness, and use a homogeneous integrating factor to solve a sample problem.

Assess exactness of the differential equation, apply rule ii to derive an integrating factor 1/(xy), transform to an exact form, then obtain log(x) minus log(y) equals a constant.

Explore solving non-exact differential equations using rule iii to determine integrating factors and convert to exact form with examples.

This lecture explains linear differential equations in the form dy/dx + p(x) y = q(x), using integrating factor μ = e^{∫ p dx} to solve.

Learn to solve linear differential equations by recognizing the pattern dy/dx + p y = q, using integrating factors, and applying methods to engineering and mechanical applications.

Learn to solve linear differential equations by recognizing exact or reducible-to-exact forms, and patterns dy/dx + p(y) x = q(y) or dx/dy + p(y) x = q(y), using integrating factors.

Bernoulli's differential equation is a special case of linear equations. Divide by y^n and substitute u = 1/y^n to convert to a linear equation with an integrating factor.

Identify a Bernoulli type and transform by dividing by y^6, substitute u = 1/y^5 to obtain a linear differential equation, then re-substitute for y.

Explore how first-order differential equations model real-world problems, from finding a curve given a slope to population, radioactivity decay, Newton's cooling, circuits, motion, and GDP.

Find the equation of a curve from a given slope dy/dx by formulating and solving a differential equation. Use the integrating factor method and an initial condition.

Explain how population growth and decay follow a differential equation where the rate of change is proportional to population, yielding dp/dt = k p and p(t)=P0 e^{kt} via separation.

Model population growth with dp/dt = k p, solving p = p0 e^{kt}. Use initial and later data from town and bacteria doubling examples to predict future populations.

Explore the radioactive decay model with the differential equation, where the rate of mass loss is proportional to the current mass with a negative constant, initial conditions, and half-life concepts.

Learn Newton's law of cooling and its differential equation for temperature decay, θ = θ0 + (θ1 - θ0) e^{-kt}, with a worked example ending at 36.4° after one hour.

Determine the time to reach 30 degrees using Newton's law of cooling, solving the differential equation with initial 100 and ambient 20, yielding 60 minutes.

Explore how differential equations model electrical circuits, solving RL, RC, and RLC cases to find current using Kirchhoff's voltage law, integrating factors, initial conditions, and boundary conditions.

Explore mechanical engineering applications of differential equations by linking velocity, displacement, and acceleration to kinematical relations, Hooke's law, Newton's second law, and higher order differential equations.

Discover how differential equations model GDP growth, monetary growth, and stock prices under initial conditions. Apply the first-order equation dY/dt = kY and its exponential solution for forecasting.

The course introduces higher order differential equations after mastering first order equations and highlights Laplace transforms as a key tool for solving both orders, plus practice via the online quiz.