31 topics
Differential equations help, from separable to Laplace
Differential equations takes away the comfort of one algorithm per chapter. A separable equation, one that needs an integrating factor and one that needs Laplace all look alike on the page, and choosing wrong costs twenty minutes before you notice the method is not closing. The methods themselves are short. Classifying the equation in front of you is the skill the course is actually testing, and it is the part nobody teaches explicitly.
Where students get stuck
First-order: separable, linear, or something else — I just start integrating
Classify before you compute; it costs one line. Get the derivative alone on the left. If the right-hand side factors into a function of x times a function of y, it is separable: divide, integrate both sides, and remember the constant lives on one side only. If y and its derivative both appear to the first power and nothing else does, write it as y' + p(x)y = q(x) and multiply through by the integrating factor e^(∫p dx), which turns the left side into the derivative of a product on purpose. Plenty of equations are both and either route works. The usual stall is trying to separate something like y' = x + y, where y appears additively so it never separates — but it is linear in about ten seconds.
My guess for the particular solution collapses to 0 = the forcing term
That is not an algebra slip, it is the guess being wrong. Undetermined coefficients only works when the trial function is not already a solution of the homogeneous equation. If it is, the operator annihilates it, the left side comes out zero and you are left with an equation that has no solution. So solve the characteristic equation first, then hold your guess up against the homogeneous solutions. If it duplicates one, multiply by t; if the root was repeated and it still duplicates, multiply by t again. For y'' − 3y' + 2y = e^t the homogeneous part already contains e^t, so the guess is Ate^t. Nothing about the forcing term told you that — the characteristic roots did.
Why check exactness first instead of going straight to an integrating factor?
Because an integrating factor is the tool for making an equation exact, so it is wasted work on one that already is. Write the equation as M(x, y)dx + N(x, y)dy = 0 and compare ∂M/∂y with ∂N/∂x. If they agree, there is a potential function F with F_x = M and F_y = N: integrate M with respect to x, differentiate that result with respect to y, match it against N to recover the function of y the first integration lost, and the answer is F(x, y) = C. Only when the two partials disagree do you go looking for a factor, and the standard test is whether (M_y − N_x)/N depends on x alone, which is what makes a factor in x alone possible.
I did both halves right and still got the wrong constants
Almost always an ordering mistake. For a non-homogeneous equation the general solution is y = y_h + y_p, and although c₁ and c₂ sit inside y_h, the initial conditions have to be imposed on the whole sum after y_p is attached. Fitting the constants to the homogeneous part first and then adding the particular solution changes the value and the slope at t = 0, so both constants come out wrong even though each piece was computed correctly. This is also one reason Laplace transforms feel cleaner on an initial value problem: y(0) and y'(0) enter the algebra at the first step, as terms in the transform of the derivative, rather than being applied at the end where they are easy to misplace.
What's covered
Differential Equations topics you can work through with a tutor, generate practice on, or turn into flashcards and a study plan.
First-order equations
- Separable equations and implicit solutions
- Linear equations and integrating factors
- Exact equations and the exactness test
- Bernoulli and homogeneous substitutions
- Slope fields and Euler's method
- Existence and uniqueness for initial value problems
Modelling and qualitative behaviour
- Exponential growth and decay
- Newton's law of cooling
- Mixing and tank problems
- The logistic equation and carrying capacity
- Autonomous equations, equilibria and stability
Second-order linear equations
- The characteristic equation: distinct, repeated and complex roots
- Linear independence and the Wronskian
- Undetermined coefficients
- Variation of parameters
- Reduction of order
- Cauchy–Euler equations
Oscillations and applications
- Mass–spring systems and simple harmonic motion
- Damped motion: under-, over- and critically damped
- Forced oscillation, beats and resonance
- RLC circuits
Laplace transforms
- Transforms, inverse transforms and partial fractions
- Transforms of derivatives and solving initial value problems
- Shifting theorems
- Step functions, impulses and the Dirac delta
- Convolution
Systems, series and beyond
- Systems of first-order equations by eigenvalues and eigenvectors
- Phase plane analysis and classification of equilibria
- Series solutions about an ordinary point
- Fourier series
- Separation of variables for the heat and wave equations
Differential Equations questions
Is this ordinary differential equations, or does it cover PDEs too?
The bulk of it is ODEs — the standard second-year course sitting after calculus and linear algebra. It also covers the introductory partial differential equations most of those courses finish on: Fourier series, and separation of variables applied to the heat and wave equations. Ask for the PDE end specifically if that is the part your term ends with.
Can it tell me whether a solution I already found is right?
Yes, and this subject is unusually good for it, because a candidate solution can be verified by substituting it back into the original equation — there is no answer key needed. It will walk that check with you and, if it fails, work backwards to the line where the method went off, which is usually the integrating factor or a dropped constant.
Can it draw a slope field or a phase portrait?
The whiteboard sketches slope fields, phase lines for autonomous equations, and the trajectories of a two-by-two system as it explains what the eigenvalues mean. Those are the topics where the picture carries the argument — a saddle and a stable node are the same algebra with a sign changed, and they look nothing alike.
Can I practise just one method, like variation of parameters?
Yes. Ask for a set on one technique and you get fresh problems with full solutions. People usually use it on the two that are pure procedure — variation of parameters and Laplace with step functions — where the only way through is repetition until the bookkeeping stops slipping.
Stuck on differential equations right now?
Talk it through out loud, share your screen, and watch it worked out step by step on a whiteboard.
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