Differential Equations

Syllabus: First order equations (linear and nonlinear), higher order linear differential equations, Cauchy's and Euler's equations, methods of solution using variation of parameters, complementary function and particular integral, partial differential equations, variable separable method, initial and boundary value problems.

10 hand-picked visual resources on this topic, 4 of them little-known gems — under each card, look for See it. Every link is checked against the live site before it ships.

First-order linear

Put it in standard form, multiply by the integrating factor, then integrate once. Everything else is bookkeeping.

$\frac{dy}{dx}+P(x)y=Q(x)$
standard form
$\text{IF}=e^{\int P\,dx},\qquad y\cdot\text{IF}=\int Q\cdot\text{IF}\,dx+C$

See it

Complementary function and particular integral

For constant-coefficient equations the CF comes from the roots of the auxiliary equation. Repeated roots bring in a factor of $t$ — that is what a $(At+B)e^{\lambda t}$ form is telling you.

$\text{distinct }m_1,m_2:\ C_1e^{m_1x}+C_2e^{m_2x}$

Cauchy–Euler equation

Recognise it by the power of $x$ matching the order of the derivative. Substituting $x=e^{z}$ turns it into a constant-coefficient equation.

$x^2y''+axy'+by=0$

PDEs by trial solution

For a linear constant-coefficient PDE, substituting an exponential trial solution collapses the whole equation to an algebraic condition on the exponents — no integration needed.

$f=e^{\xi x+\eta y}\Rightarrow af_{xx}+bf_{yy}=f\ \text{gives}\ a\xi^2+b\eta^2=1$

Wave equation and d'Alembert

Any function of $(x-ct)$ or $(x+ct)$ solves it — a shape travelling right or left without changing. Checking an option reduces to reading off its propagation speed.

$\frac{\partial^2f}{\partial t^2}=c^2\frac{\partial^2f}{\partial x^2}$

Below: on the syllabus, but no past question for it in this bank.

Variation of parameters, and the first-order forms
syllabus

Use variation of parameters when the right-hand side is not one of the shapes the method of undetermined coefficients handles — $\tan x$ , $\sec x$ , $1/x$ . For first-order equations, check separable first, then exact, then Bernoulli; only then reach for the integrating factor.

$y_p=-y_1\!\int\!\frac{y_2r}{W}dx+y_2\!\int\!\frac{y_1r}{W}dx$

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Differential Equations

10 hand-picked visual resources on this topic, 4 of them little-known gems — under each card, look for See it. Every link is checked against the live site before it ships.

First-order linear

Put it in standard form, multiply by the integrating factor, then integrate once. Everything else is bookkeeping.

$\frac{dy}{dx}+P(x)y=Q(x)$
standard form
$\text{IF}=e^{\int P\,dx},\qquad y\cdot\text{IF}=\int Q\cdot\text{IF}\,dx+C$

See it

Complementary function and particular integral

For constant-coefficient equations the CF comes from the roots of the auxiliary equation. Repeated roots bring in a factor of $t$ — that is what a $(At+B)e^{\lambda t}$ form is telling you.

$\text{distinct }m_1,m_2:\ C_1e^{m_1x}+C_2e^{m_2x}$

Cauchy–Euler equation

Recognise it by the power of $x$ matching the order of the derivative. Substituting $x=e^{z}$ turns it into a constant-coefficient equation.

$x^2y''+axy'+by=0$

PDEs by trial solution

For a linear constant-coefficient PDE, substituting an exponential trial solution collapses the whole equation to an algebraic condition on the exponents — no integration needed.

$f=e^{\xi x+\eta y}\Rightarrow af_{xx}+bf_{yy}=f\ \text{gives}\ a\xi^2+b\eta^2=1$

Wave equation and d'Alembert

Any function of $(x-ct)$ or $(x+ct)$ solves it — a shape travelling right or left without changing. Checking an option reduces to reading off its propagation speed.

$\frac{\partial^2f}{\partial t^2}=c^2\frac{\partial^2f}{\partial x^2}$

Below: on the syllabus, but no past question for it in this bank.

Variation of parameters, and the first-order forms
syllabus

$y_p=-y_1\!\int\!\frac{y_2r}{W}dx+y_2\!\int\!\frac{y_1r}{W}dx$

Differential Equations

First-order linear

Complementary function and particular integral

Cauchy–Euler equation

PDEs by trial solution

Wave equation and d'Alembert

Variation of parameters, and the first-order formssyllabus

Differential Equations

First-order linear

Complementary function and particular integral

Cauchy–Euler equation

PDEs by trial solution

Wave equation and d'Alembert

Variation of parameters, and the first-order formssyllabus

Variation of parameters, and the first-order forms
syllabus

Variation of parameters, and the first-order forms
syllabus