Break What? Forum

What is Calculus II · Differential Equations

a Differential Equation?

An equation involving a function and its derivatives — the language of change in science, engineering, and nature.

Simple Definition

A differential equation is an equation whose unknown is a function rather than a number, and which involves that function together with one or more of its derivatives. Solving a differential equation means finding the function (or family of functions) that satisfies the relationship.

Unlike algebraic equations like $x^2 = 4$, a differential equation describes how something changes — making it the natural language for physics, biology, economics, and engineering.

General form — 1st order ODE

$$\frac{dy}{dx} = f(x,\, y)$$

General form — 2nd order ODE

$$a\,y'' + b\,y' + c\,y = g(x)$$

The order of a differential equation is the highest derivative that appears. The degree is the power to which that highest derivative is raised (when the equation is polynomial in derivatives).

Intuition

🌡️

Newton's law of cooling

A hot cup of coffee cools faster when it's much hotter than the room and slower as temperatures equalise. This rate-of-change relationship — "the derivative of temperature is proportional to the difference in temperatures" — is a differential equation:

$\dfrac{dT}{dt} = -k(T - T_{\text{room}})$

Solving it tells you the temperature at every future moment. The equation doesn't give you $T$ directly — it describes how $T$ changes, and integration uncovers $T$ itself.

🔄

A rule about change, not a value

An algebraic equation pins down a number. A differential equation pins down a pattern of change. There are usually infinitely many functions satisfying a differential equation — one for each initial condition (starting value) you impose. That's why solutions come with arbitrary constants $C$.

Classification

By variables

ODE vs PDE

Ordinary (ODE): one independent variable. Partial (PDE): two or more. Most Calc II content is ODEs.

ODE: $y' = 2y$
PDE: $\frac{\partial u}{\partial t} = k\frac{\partial^2 u}{\partial x^2}$

By order

1st, 2nd, … order

Order = highest derivative present. First-order equations model simple growth, cooling, mixing. Second-order model oscillations and circuits.

1st order: $y' + y = 0$
2nd order: $y'' + \omega^2 y = 0$

By linearity

Linear vs Nonlinear

Linear: $y$ and all its derivatives appear to the first power only, no products of $y$ with its derivatives. Otherwise nonlinear.

Linear: $y'' - 3y' + 2y = e^x$
Nonlinear: $y' = y^2$

By right-hand side

Homogeneous vs Non-homogeneous

A linear ODE is homogeneous if $g(x) = 0$, otherwise non-homogeneous. The solution structure differs in each case.

Hom: $y'' + y = 0$
Non-hom: $y'' + y = \sin x$

Solving Separable Equations

A separable first-order ODE can be written so that all $y$-terms are on one side and all $x$-terms on the other. Then both sides can be integrated independently.

Separable form

$$\frac{dy}{dx} = g(x)\cdot h(y) \quad \implies \quad \frac{dy}{h(y)} = g(x)\,dx$$

Separate variables

Rewrite so all $y$ (and $dy$) are on the left, all $x$ (and $dx$) on the right.

$\dfrac{dy}{h(y)} = g(x)\,dx$

Integrate both sides

Integrate independently. Add a single constant $C$ on one side.

$\displaystyle\int \frac{dy}{h(y)} = \int g(x)\,dx + C$

Solve for $y$

Isolate $y$ to get the general solution. Apply an initial condition $y(x_0) = y_0$ to find $C$ for the particular solution.

Where They Appear

📈

Biology / Finance

Exponential Growth & Decay

$\dfrac{dP}{dt} = kP$
Solution: $P = P_0 e^{kt}$

☕

Physics / Engineering

Newton's Law of Cooling

$\dfrac{dT}{dt} = -k(T - T_\infty)$
Solution: $T = T_\infty + Ce^{-kt}$

🔋

Electrical Engineering

RC Circuit

$R\,\dfrac{dq}{dt} + \dfrac{q}{C} = V(t)$
Charge as a function of time

🌊

Physics

Simple Harmonic Motion

$y'' + \omega^2 y = 0$
Solution: $y = A\cos\omega t + B\sin\omega t$

🧪

Chemistry

Mixing Problems

$\dfrac{dA}{dt} = \text{rate in} - \text{rate out}$
Amount of substance over time

🦠

Epidemiology

Logistic Growth

$\dfrac{dP}{dt} = rP\!\left(1-\dfrac{P}{K}\right)$
Population with carrying capacity

Worked Examples

Example 1 — Separable

Solve $\dfrac{dy}{dx} = xy$, with $y(0) = 2$.

Separate: $\dfrac{dy}{y} = x\,dx$

Integrate: $\ln|y| = \dfrac{x^2}{2} + C_1$

Exponentiate: $y = Ce^{x^2/2}$

Apply $y(0)=2$: $C = 2$

Particular solution: $y = 2e^{x^2/2}$

Example 2 — Exponential decay

A substance decays at rate proportional to its mass. Half-life is 10 years. Find mass at $t = 30$.

$\dfrac{dm}{dt} = -km$ → $m = m_0 e^{-kt}$

Half-life: $\dfrac{m_0}{2} = m_0 e^{-10k}$
$\Rightarrow k = \dfrac{\ln 2}{10}$

At $t = 30$: $m = m_0 e^{-3\ln 2} = \dfrac{m_0}{8}$

After 3 half-lives, $\tfrac{1}{8}$ of the original mass remains.

Example 3 — Linear 1st order

Solve $y' + 2y = 4x$.

Integrating factor: $\mu = e^{\int 2\,dx} = e^{2x}$

Multiply: $(e^{2x}y)' = 4xe^{2x}$

Integrate (by parts):
$e^{2x}y = e^{2x}(2x-1) + C$

General solution: $y = 2x - 1 + Ce^{-2x}$

Example 4 — 2nd order homogeneous

Solve $y'' - 5y' + 6y = 0$.

Characteristic equation:
$r^2 - 5r + 6 = 0$
$(r-2)(r-3) = 0$
$r_1 = 2,\quad r_2 = 3$

Two distinct real roots:

General solution: $y = C_1 e^{2x} + C_2 e^{3x}$

Common Mistakes

✗ Forgetting the constant of integration

Every integration step introduces a $+C$. In a differential equation, this constant represents the entire family of solutions. Dropping it gives only one particular solution, not the general one.

✗ Dividing by zero when separating

Dividing both sides by $h(y)$ to separate variables is only valid when $h(y) \ne 0$. Constant solutions where $h(y) = 0$ (called equilibrium solutions) must be checked separately.

✗ Misidentifying order and degree

Order is the highest derivative present. Degree is the power of that highest derivative (when rational). In $(y'')^3 + y = 0$, the order is 2 and the degree is 3 — not the other way around.

✗ Using the wrong solution method

Separable, linear, exact, and homogeneous equations each require different techniques. Always classify first. Trying to separate a non-separable equation wastes time and produces wrong answers.

Related Concepts

Ready to test your knowledge?

Differential equations appear in every branch of science and engineering. Practice setting them up and solving them with our quizzes.

Browse All Quizzes →