Ordinary differential equations

• What Is the Solvability Condition for a Differential Operator?
  For a Fredholm differential operator, Lu = f is solvable exactly when f is orthogonal to every solution of the adjoint homogeneous equation.
• How to Solve $2\sqrt{xy}\,\frac{dy}{dx}=1$
  Separate the variables, integrate carefully, and state the real-domain branches for 2√(xy)y′ = 1, including the usual positive-x, positive-y solution.
• How Do You Create a Second-Order Equation from a Spring Constant?
  Use Hooke’s law and Newton’s second law to build a spring’s second-order equation, then see how mass and spring constant determine its motion.
• How to Write Coupled Nonlinear ODEs in State-Space Form
  Convert coupled nonlinear ODEs to first-order state-space form by choosing state variables, solving for their derivatives, and defining outputs.
• How to Model Salt Dissolving in a Well-Mixed Tank
  Model salt in a well-mixed tank with a differential equation, separate variables, and find the salt amount over time from flow rates.
• How to Show That a Function Solves a Differential Equation
  To verify that a function solves a differential equation, compute its derivatives and substitute them into the equation and any initial conditions.
• How to Find the Rate of Change of Resistance
  Find resistance’s rate of change by differentiating its model with respect to time, temperature, or another changing quantity, then substitute values.
• Can You Prove Sensitivity to Initial Conditions Numerically?
  Numerical simulations can provide evidence of sensitivity to initial conditions, but a plot alone is not a proof. Here is a reliable test using the Lorenz system.
• What Differential Equation Describes Newtonian Motion?
  Newtonian motion is governed by force equals mass times acceleration; a force law and initial position and velocity determine the resulting trajectory.
• How to Find a Lyapunov Function for a Nonlinear Time-Varying ODE
  For a time-varying nonlinear ODE, test stability with a positive Lyapunov function and its full derivative, including explicit time dependence.
• How to Model Projectile Motion with Differential Equations
  Build a projectile-motion model from Newton’s law, solve its differential equations, and calculate flight time, maximum height, and range.
• How to Find a New Output When the Input Changes
  For a linear time-invariant system, find the new output by multiplying its transfer function by the new input transform, then invert; account for initial conditions.
• How to Find the Inverse Laplace Transform of 1/[s²(s² + ω²)]
  Find the inverse Laplace transform of 1/[s²(s² + ω²)] by partial fractions, and check the result and the special case ω = 0.
• How to Solve a First-Order ODE with an Integrating Factor
  Solve a first-order linear differential equation by finding its integrating factor, turning the left side into a product derivative, and integrating.
• How to Calculate Barometric Pressure at an Altitude
  Use hydrostatic balance and the ideal-gas law to derive the exponential barometric formula and estimate pressure at a chosen altitude.
• How to Derive the Hermite Polynomial Derivative Relation
  Differentiate the physicists’ Hermite equation to show H_n' solves the degree-(n−1) equation, then compare leading coefficients to get H_n'=2nH_{n−1}.
• How to Use Variation of Parameters for a 3×3 System
  Use a fundamental matrix to solve a nonhomogeneous 3×3 linear ODE system by variation of parameters, with a worked example.