Problem-solving recipes

A step-by-step procedure for every problem type this course teaches — what to do first, what to do next, and how to tell which case you're in.

What are you being asked to do?

Start with what kind of equation — or what question about one — you're facing.

Given a 2×2 matrix (readiness review, or a linear system further on) — the eigenvalue/eigenvector recipe.

Given an initial-value problem, need the solution and where it's valid — solve it, then find the interval of validity from where the solution itself breaks down.

First-order equation, need to solve it — check separability first (cheapest test, pure algebra); if not separable, check whether it's linear in standard form; a $y^n$ term on the right instead means Bernoulli; $y'=F(y/x)$ means homogeneous (the substitution kind, not the RHS-is-zero kind); an equation that isn't exact needs an integrating factor first. See Choosing a solution method for the full decision order. A rate-in/rate-out word problem (tank, mixing) usually just IS one of these once you set it up.

Given an autonomous family with a parameter, need equilibria or a bifurcation — the equilibria recipe first; the bifurcation-value recipe once you need the exact boundary between qualitatively different behaviors.

Given a numerical-methods error plot or step size, need to read accuracy or stability off it — the order-of-accuracy or Euler- stability-threshold recipes.

A first-order modeling word problem (drug elimination, reaction rate, falling object with drag) — these usually reduce to one of the first-order recipes above; the modeling-specific recipes below add the extra "solve for the requested real-world quantity" step on top.

Given two solutions of a 2nd-order linear equation, need independence or the general solution — the Wronskian recipe, then (once independence is confirmed) the general-solution recipe; Abel's formula gives $W$ without ever computing the two solutions' derivatives directly.

Second-order, homogeneous, constant-coefficient ( $ay''+by'+cy=0$ ) — one recipe covers all three characteristic-root cases (distinct real, repeated, complex). The Cauchy-Euler equation is the same idea with a different characteristic equation — its own recipe covers all its root cases too.

A mass-spring or RLC vibration problem — Hooke's law recipe to set up the equation, then the damping-classification recipe (covers overdamped, critically damped, and underdamped in one chain) for free vibration; the undamped case is its own simpler recipe.

Second-order, nonhomogeneous ( $ay''+by'+cy=g(x)$ ) — undetermined coefficients when $g(x)$ is a polynomial, exponential, or sine/cosine (with the resonance fix if the naive guess collides with the homogeneous solution); variation of parameters otherwise. Amplitude response and beats are specific downstream questions about a sinusoidally-forced system's steady-state behavior, not alternate ways to find the particular solution itself.

Higher than second order — the same characteristic-equation idea, one recipe for real roots; variation of parameters generalizes too, at the cost of a bigger linear system to solve for the unknown functions.

Given a function, need its Laplace transform, or given $Y(s)$ , need the IVP it solves — the from-the-definition recipe for a first transform; convolution for a product of transforms; impulse forcing for a $\delta$ -function right-hand side.

Readiness — eigenvalues of a 2×2 matrix

Find the eigenvalues and eigenvectors of a 2×2 matrix

Use when: you have a $2\times2$ matrix $A$ — often the coefficient matrix of a system $\mathbf x'=A\mathbf x$ — and need every $\lambda$ and matching $\mathbf v$ satisfying $A\mathbf v=\lambda\mathbf v$

Goal: every eigenvalue of $A$ , each paired with an eigenvector spanning its eigenspace

Eigenvalues and eigenvectors of a 2x2 matrix

Initial-value problems

Solve an initial-value problem and find its interval of validity

Use when: you're given a differential equation together with a starting value $y(x_0)=y_0$ and asked for the particular solution and how far it can be trusted

Goal: the particular solution passing through $(x_0,y_0)$ , plus the largest interval around $x_0$ over which it's actually valid

Initial-value problems

Solving a first-order equation

Solve a separable first-order equation

Use when: the equation can be written as $\dfrac{dy}{dx}=f(x)g(y)$ — every $y$ -term collects on one side, every $x$ -term on the other

Goal: an implicit or explicit solution for $y$ , plus any equilibrium solutions the separation step would otherwise lose

Separable equations

Try this first — it's the cheapest test to run.

Solve a first-order linear equation

Use when: the equation is (or can be rearranged into) $\dfrac{dy}{dx}+P(x)y=Q(x)$ — $y$ and $y'$ appear only to the first power, never multiplied together or raised to a power

Goal: an explicit solution $y(x)$

First-order linear equations and integrating factors

Solve a Bernoulli equation

Use when: the equation has the form $y'+p(x)y=q(x)y^n$ with $n\ne0,1$

Goal: a linear equation in $v$ you already know how to solve, then the answer converted back to $y$

The Bernoulli equation

A first-order linear equation once you divide by $y^n$ and substitute.

Solve a homogeneous first-order equation

Use when: $\dfrac{dy}{dx}=f(x,y)$ , and dividing the numerator and denominator by the highest power of $x$ present leaves an expression depending only on the ratio $y/x$

Goal: an implicit or explicit solution for $y$ , converted back from the substituted variable $v$

Homogeneous first-order equations

Recognized by $y'=F(y/x)$ ; solved with the substitution $v=y/x$ , not to be confused with a homogeneous (RHS-zero) linear equation.

Find and apply an integrating factor for a non-exact equation

Use when: an equation $M\,dx+N\,dy=0$ fails the exactness test ( $M_y\ne N_x$ ), and at least one of the two special-case ratios below simplifies to a function of a single variable

Goal: a factor $\mu$ that restores exactness, so the exact-equation method solves the repaired equation

Integrating factors for non-exact equations

Build and solve a linear rate-in/rate-out model

Use when: a word problem describes a quantity $y$ that increases by an incoming rate and decreases by a rate proportional to $y$ itself — a tank, a circuit, an account balance

Goal: the standard-form linear equation for $y$ , then $y(t)$ itself via the integrating-factor method

Applications of the integrating-factor method

The tank/mixing word-problem version of the linear-equation recipe above.

Equilibria and bifurcation of an autonomous family

Find the equilibria of a harvested logistic population

Use when: the equation has the form $\dfrac{dy}{dt}=ry(1-y/K)-h$ — logistic growth with a constant harvest rate $h$ subtracted

Goal: both equilibria (when they exist), which one is stable, and the survival threshold below which the model predicts extinction

Logistic growth with harvesting · extended later in this course

Find the bifurcation value of a one-parameter family

Use when: you have a family of autonomous equations $\dfrac{dy}{dt}=f(y,c)$ depending on a parameter $c$ , and want the exact value of $c$ where the number of equilibria changes

Goal: the bifurcation value $c^\star$ , plus what happens to the equilibria on each side of it

Bifurcation of a 1-D autonomous family

Extends the equilibria recipe above with the exact parameter value where two equilibria merge.

Numerical methods — reading accuracy and stability

Read a method's order of accuracy off two points on an error plot

Use when: you have (or can read off a log-log error-versus-step-size plot) two pairs $(h_1,E_1)$ and $(h_2,E_2)$ for the same method on the same problem

Goal: the numerical order of accuracy $p$ , with a check that the two points actually sit on the straight, trustworthy part of the plot

Reading order of accuracy off an error plot

Find Euler's stability threshold for a stiff equation

Use when: the equation locally reduces to $y'=-k(y-g(x))$ (or exactly $y'=-ky$ ) near an equilibrium, with $k$ large and $g(x)$ slowly varying, and you need the largest Euler step size that stays stable

Goal: the stability bound $h<2/k$ , plus a way to check whether a specific candidate $h$ satisfies it

Stiffness, informally

Modeling with a first-order equation

Find when a decaying drug concentration reaches a given level

Use when: you know $C_0$ , the elimination constant $k_e$ , and a target concentration — including the special target $C_0/2$ , the half-life

Goal: the exact time $t$ at which $C(t)$ first reaches that target level

Pharmacokinetics: a drug-elimination model

Solve a reaction rate law $-d[A]/dt=k[A]^n$ for $[A](t)$

Use when: the rate law is $-\dfrac{d[A]}{dt}=k[A]^n$ for some order $n$ , and you know $[A](0)=[A]_0$

Goal: an explicit formula for $[A](t)$ , plus (if needed) whether the half-life depends on the starting concentration

Chemical reaction rates

Find terminal velocity and predict how a falling object approaches it

Use when: you have an autonomous velocity equation $\dfrac{dv}{dt}=g-\dfrac{k}{m}v$ (or any equation of the form $v'=(\text{constant})-(\text{positive constant})v$ ) and an initial speed $v_0$

Goal: the terminal velocity $v^\star$ , confirmation that it's stable, and which direction a given $v_0$ approaches it from

Terminal velocity with air resistance

Independence and the general solution of a 2nd-order linear equation

Check two solutions for linear independence with the Wronskian

Use when: you have two solutions $y_1,y_2$ of the same equation $y''+p(x)y'+q(x)y=0$ and need to know whether they're independent

Goal: a verdict on independence, valid across the whole interval, from one computation

The Wronskian

Find W(x) with Abel's formula

Use when: you know a linear homogeneous equation's coefficient $p(x)$ and one value $W(x_0)$ , but not $y_1,y_2$ themselves

Goal: an explicit formula for $W(x)$ on the whole interval, without ever finding $y_1,y_2$

Abel's formula

Gets $W$ directly from the equation's own coefficient, without differentiating either solution.

Write and use the general solution of a linear homogeneous equation

Use when: you have two solutions $y_1,y_2$ of $y''+p(x)y'+q(x)y=0$ and need the complete solution set, possibly pinned down by initial conditions

Goal: the general solution $y=C_1y_1+C_2y_2$ , and — if conditions are given — the one specific solution matching them

The general solution of a linear homogeneous equation

Needs a confirmed fundamental set from the Wronskian recipe above first.

Solving a homogeneous second-order linear equation

Solve a homogeneous second-order linear ODE

Use when: you have any equation of the form $ay''+by'+cy=0$ with constant coefficients $a,b,c$

Goal: the general, real-valued solution $y(x)$ , covering every possible characteristic-root outcome

Complex roots and Euler's formula

Covers distinct real, repeated, and complex root cases in one recipe.

Solving a Cauchy-Euler equation

Solve a Cauchy-Euler equation

Use when: you have $ax^2y''+bxy'+cy=0$ ( $x>0$ ), and the characteristic equation's roots might be repeated or complex, not just distinct and real

Goal: the general, real-valued solution $y(x)$ , covering every possible root outcome

Cauchy-Euler with repeated and complex roots

Covers distinct real, repeated, and complex root cases in one recipe — the Cauchy-Euler analog of the constant-coefficient recipe above.

Mechanical and electrical vibrations

Apply Hooke's law to a spring problem

Use when: you're given two of the three quantities in $F=-kx$ (the spring constant $k$ , the displacement $x$ , or the restoring force $F$ ) and asked for the third

Goal: the missing quantity in $F=-kx$ , with its sign correctly read as a real physical direction

Mass-spring systems and Hooke's law

Sets up the equation of motion before any of the recipes below run.

Solve an undamped free-vibration IVP

Use when: you have $mx''+kx=0$ (no damping term at all) together with initial conditions $x(0)=x_0$ , $x'(0)=v_0$

Goal: the position function $x(t)$ as a fixed-amplitude sinusoid, with $C_1,C_2$ pinned down by the initial conditions

Undamped free vibration, natural frequency

Solve a mass-spring free-vibration IVP

Use when: you have $mx''+cx'+kx=0$ with initial conditions $x(0)=x_0$ , $x'(0)=v_0$ , covering every possible level of damping

Goal: $x(t)$ for all three damping regimes, found by classifying the discriminant (or the damping ratio) first

Underdamping and the damping ratio

Covers overdamped, critically damped, and underdamped free vibration in one recipe chain.

Finding a particular solution for a nonhomogeneous equation

Find a particular solution with undetermined coefficients

Use when: you have a nonhomogeneous linear ODE $ay''+by'+cy=g(x)$ where $g(x)$ is a polynomial, exponential, sine/cosine, or a sum/product of these

Goal: a particular solution $y_p$ to add to the homogeneous solution $y_h$

The method of undetermined coefficients · extended later in this course

Use when g(x) is a polynomial, exponential, or sine/cosine.

Find a particular solution with undetermined coefficients

Use when: you have a nonhomogeneous linear ODE $ay''+by'+cy=g(x)$ , including the case where the naive trial function duplicates a term already in $y_c$

Goal: a particular solution $y_p$ that actually works, even when the naive guess collides with the homogeneous solution

The resonance modification rule

The same method, extended for when the naive guess collides with the homogeneous solution.

Apply variation of parameters to a nonhomogeneous equation

Use when: your forcing term $g(x)$ doesn't fit undetermined coefficients' polynomial, exponential, or sine-cosine trial forms, or you just need a method that works for any $g(x)$

Goal: a particular solution $y_p=u_1y_1+u_2y_2$ , built entirely from the homogeneous fundamental set and two integrals

Deriving the variation-of-parameters formula

Use when g(x) doesn't fit undetermined coefficients' guessable forms.

Find the amplitude of a driven system's steady-state response

Use when: you have a forced equation $mx''+cx'+kx=F_0\cos(\omega t)$ and need the size of the steady-state oscillation at one specific driving frequency $\omega$

Goal: the number $X(\omega)$ , the steady-state amplitude at that driving frequency, plus whether it sits near the resonance peak

Amplitude response and resonance

A downstream question about a sinusoidally-driven system's steady-state size, not an alternate way to find the particular solution.

Write the beat solution for an undamped near-resonant drive

Use when: you have an undamped forced equation $x''+\omega_0^2x=F_0\cos(\omega t)$ with rest initial conditions and a driving frequency $\omega$ close to, but not equal to, the natural frequency $\omega_0$

Goal: the full solution $x(t)$ written in envelope form, with the slow envelope frequency and fast carrier frequency identified separately

Beats

The undamped, near-(but not exactly)-resonant special case of the amplitude-response question above.

Higher-order linear equations

Solve a higher-order homogeneous equation with real roots

Use when: you have a constant-coefficient homogeneous equation $a_ny^{(n)}+\cdots+a_1y'+a_0y=0$ of order 3 or higher, and its characteristic polynomial factors entirely into real roots

Goal: the general solution $y(x)$ , built as one linear combination of $n$ independent functions, one set per root

The characteristic polynomial and its roots

Apply variation of parameters to a higher-order equation

Use when: you have a nonhomogeneous linear equation of order 3 or higher, already in standard form, with a known fundamental set $y_1,\ldots,y_n$ for the homogeneous equation

Goal: a particular solution $y_p=u_1y_1+\cdots+u_ny_n$ , found by solving one $n\times n$ linear system and integrating

Variation of parameters at higher order

The same formula as the 2nd-order version, generalized to a bigger linear system for the unknown functions.

The Laplace transform

Compute a Laplace transform directly from the definition

Use when: you need $\mathcal{L}\{f(t)\}(s)$ for an $f(t)$ not yet in a lookup table, or you're deriving or verifying a table entry

Goal: a closed-form $F(s)$ , together with the values of $s$ for which the improper integral actually converges

Definition and motivation

Compute a convolution directly from the integral

Use when: you need $(f*g)(t)$ and don't yet have $F(s)$ and $G(s)$ both in a form you can simply multiply and invert

Goal: a closed-form expression for $(f*g)(t)$ as an ordinary function of $t$

Convolution and the convolution theorem

Solve an IVP with impulse forcing for Y(s)

Use when: the forcing term $g(t)$ in $ay''+by'+cy=g(t)$ contains a Dirac delta $\delta(t-t_0)$ , alone or added to other forcing

Goal: the algebraic equation solved for $Y(s)$ , with every delta term correctly transformed, ready for whichever inversion technique fits its shape

The Dirac delta and impulse forcing

Sets up $Y(s)$ from a $\delta$ -function forcing term; inverting back to $y(t)$ uses the transform table, not this recipe.

Problem-solving recipes

Readiness — eigenvalues of a 2×2 matrix

Find the eigenvalues and eigenvectors of a 2×2 matrix

Initial-value problems

Solve an initial-value problem and find its interval of validity

Solving a first-order equation

Solve a separable first-order equation

Solve a first-order linear equation

Solve a Bernoulli equation

Solve a homogeneous first-order equation

Find and apply an integrating factor for a non-exact equation

Build and solve a linear rate-in/rate-out model

Equilibria and bifurcation of an autonomous family

Find the equilibria of a harvested logistic population

Find the bifurcation value of a one-parameter family

Numerical methods — reading accuracy and stability

Read a method's order of accuracy off two points on an error plot

Find Euler's stability threshold for a stiff equation

Modeling with a first-order equation

Find when a decaying drug concentration reaches a given level

Solve a reaction rate law -d[A]/dt=k[A]^n for [A](t)

Find terminal velocity and predict how a falling object approaches it

Independence and the general solution of a 2nd-order linear equation

Check two solutions for linear independence with the Wronskian

Find W(x) with Abel's formula

Write and use the general solution of a linear homogeneous equation

Solving a homogeneous second-order linear equation

Solve a homogeneous second-order linear ODE

Solving a Cauchy-Euler equation

Solve a Cauchy-Euler equation

Mechanical and electrical vibrations

Apply Hooke's law to a spring problem

Solve an undamped free-vibration IVP

Solve a mass-spring free-vibration IVP

Finding a particular solution for a nonhomogeneous equation

Find a particular solution with undetermined coefficients

Find a particular solution with undetermined coefficients

Apply variation of parameters to a nonhomogeneous equation

Find the amplitude of a driven system's steady-state response

Write the beat solution for an undamped near-resonant drive

Higher-order linear equations

Solve a higher-order homogeneous equation with real roots

Apply variation of parameters to a higher-order equation

The Laplace transform

Compute a Laplace transform directly from the definition

Compute a convolution directly from the integral

Solve an IVP with impulse forcing for Y(s)

Solve a reaction rate law $-d[A]/dt=k[A]^n$ for $[A](t)$