Differential Equations
26 generated practice sets. Every one writes a new problem with new numbers each time you open it, scores each part of your answer separately, and shows the full worked solution.
1.1Introduction to ODEs & Direction FieldsRead an autonomous equation without solving it: equilibria, a sign chart, stability, and slopes straight off the direction field.1.2Solutions to Some ODEsThe constant-coefficient first-order template, solved once as a constant plus an exponential, and what the sign of the rate decides about the long run.1.3Classification of Differential EquationsOrder, linear versus nonlinear, and the substitutions that turn a proposed solution into a characteristic polynomial.2.1First-Order Linear EquationsIntegrating factors in the right order: standard form first, then the factor, then the product rule, then one integration.2.2Separable EquationsSeparate, integrate, apply the initial condition early, pick the right branch, and state the interval of validity.2.3Modeling with First-Order ODEsMixing tanks, Newton's law of cooling and falling bodies with drag — rate in minus rate out before any calculus.2.4Existence & Uniqueness (first order)When the theorem guarantees a unique solution near a point, and what its silence does and does not mean.2.5Population DynamicsLogistic growth, threshold models and constant harvesting, all read off the shape of one parabola.2.7Euler MethodsStepping along the direction field by hand, the improved Euler correction, and why global error is one order worse than local.3.1Linear Homogeneous 2nd-Order ODEsCharacteristic roots, real and distinct, with initial conditions — and reading long-run behavior off the largest root.3.2Linear Independence & the WronskianWronskians, fundamental sets, Abel's theorem, and finding the largest interval on which a solution is guaranteed.3.3Complex RootsNegative discriminant, Euler's formula, and the split between the decay rate and the frequency.3.4Repeated RootsOne root, two solutions — where the extra factor of t comes from and how it changes the initial-condition algebra.3.5Method of Undetermined CoefficientsPick the right trial form, repair it when it collides with the homogeneous solution, then solve for the constants.3.7Free Mechanical VibrationsMass, spring and damper: natural frequency from a hanging stretch, amplitude and phase, and the three damping cases.3.8Forced Mechanical VibrationsTransient versus steady state, the amplitude formula, beats, and why resonance needs zero damping.6.1Laplace TransformsWorking the transform table instead of the definition, plus the first shifting theorem.6.2Solving IVPs with Laplace TransformsTransform the IVP, split with partial fractions or complete the square, and read the answer off the table.6.3Step FunctionsThe unit step as a switch: writing piecewise forcing in step form, and the second shifting theorem both ways.6.4Discontinuous Forcing FunctionsIVPs whose forcing switches on or off, solved in one pass instead of interval by interval.7.1Linear Systems of ODEsTurning a high-order equation into a first-order system, and reading the companion matrix back again.7.2Linear Algebra ReviewDeterminants, invertibility and independence — the six equivalent statements that eigenvalue work keeps using.7.3Eigenvalues and EigenvectorsTrace and determinant to the characteristic polynomial, and what the sign pattern says about the phase portrait.7.5Homogeneous Linear SystemsEigenvalues and eigenvectors to the general solution, initial conditions, and what the phase portrait looks like.7.6Complex EigenvaluesComplex eigenvalues, real solutions from Euler's formula, and the spiral sink / source / centre split.7.7Repeated EigenvaluesOne eigenvalue, one eigenvector: generalised eigenvectors and the improper node.