Differential Equations Lecture Notes ✦

Techniques for solving equations involving the first derivative.

Solve ( \frac{dy}{dx} + 2y = e^{-x} ). Integrating factor : ( \mu(x) = e^{\int 2,dx} = e^{2x} ). Multiply through: ( e^{2x}y' + 2e^{2x}y = e^{x} ) Left side is ( \frac{d}{dx}(e^{2x}y) = e^{x} ) Integrate: ( e^{2x}y = e^{x} + C ) Thus ( y = e^{-x} + Ce^{-2x} ).

A differential equation (DE) is any equation that contains at least one derivative of an unknown function. Our goal is to find the function itself. differential equations lecture notes

A visual way to understand the stability of the system (e.g., nodes, saddles, and spirals). 💡 Key Takeaways for Success

Students should read the relevant section attending lecture, then use the worked examples as a reference while solving homework. Instructors are free to adapt the notes (under CC BY-NC 4.0 license) to fit their course pacing. Multiply through: ( e^{2x}y' + 2e^{2x}y = e^{x}

If you find any errors in the derivations or have specific questions about the methods, drop a comment below!

consisting of polynomials, exponentials, or sines/cosines. You "guess" the form of A visual way to understand the stability of the system (e

The heart of classical mechanics and circuit theory.

Second-order equations are vital for studying oscillations, such as springs and circuits. The standard form is Homogeneous Equations ( We assume a solution of the form and solve the : Distinct Real Roots: Repeated Roots: Complex Roots ( ): Non-Homogeneous Equations ( The general solution is is the homogeneous solution and is a particular solution. Undetermined Coefficients: Best for

Involve functions of multiple independent variables and their partial derivatives.