Calculus is the mathematics of change, and Mathematics – I is where most of us first used it to describe real systems. Differential equations describe how a quantity changes in terms of itself: a cooling server, a charging capacitor, a growing queue. This article revisits the five units of the JNTUK course in short form, with the method and the meaning of each.
Unit 1: Differential equations of first order
A differential equation relates a function to its derivatives. The order is the highest derivative; the degree is the power of that derivative. A solution contains arbitrary constants, and an initial condition fixes them.
Methods for first-order equations
| Type | Form | Method |
|---|---|---|
| Variables separable | dy/dx = f(x) g(y) | Move all y terms to one side, all x terms to the other, integrate both |
| Homogeneous | dy/dx = f(y/x) | Substitute y = vx to make it separable |
| Exact | M dx + N dy = 0 with ∂M/∂y = ∂N/∂x | Solution is a function F(x, y) = C, found by integrating M and N |
| Linear | dy/dx + P y = Q | Multiply by the integrating factor e∫P dx, then integrate |
| Bernoulli | dy/dx + P y = Q yⁿ | Substitute z = y1−n to make it linear |
If an equation is not exact, an integrating factor can sometimes make it exact. A small example: dy/dx = ky separates to dy/y = k dx, giving y = Cekx, the exponential growth or decay law.
Applications
- Newton’s law of cooling: dT/dt = −k (T − T0). The temperature approaches the surroundings exponentially.
- Growth and decay: population, radioactive decay, compound interest.
- Orthogonal trajectories: curves that cross a given family at right angles.
- Electric circuits: a series RL or RC circuit gives a first-order linear equation for current or voltage.
Today: exponential behaviour is everywhere in operations. Retry back-off, cache expiry and the decay of a moving average are all e−kt curves, and the “half-life” of a metric is the same idea as radioactive decay.
Unit 2: Linear differential equations of higher order
The general form is an y(n) + … + a1 y′ + a0 y = f(x). The complete solution is:
y = yc + yp, the complementary function plus a particular integral.
Complementary function (f(x) = 0)
Write the auxiliary equation by replacing y(k) with mk, then solve for m.
| Roots of the auxiliary equation | Complementary function |
|---|---|
| Real, distinct m₁, m₂ | C₁em₁x + C₂em₂x |
| Real, repeated m | (C₁ + C₂x) emx |
| Complex α ± iβ | eαx (C₁ cos βx + C₂ sin βx) |
Particular integral
| Right-hand side f(x) | Short method |
|---|---|
| eax | Replace D by a (if that makes the denominator zero, multiply by x) |
| sin ax or cos ax | Replace D² by −a² |
| Polynomial xⁿ | Expand 1 / f(D) as a series in D |
| eax V(x) | Move eax out and replace D by D + a |
Here D means d/dx. For general cases there are two more tools: the method of variation of parameters (works for any f(x) once you know yc) and the Cauchy-Euler equation, solved by substituting x = et.
Applications
A mass on a spring, y″ + (c/m) y′ + (k/m) y = 0, models free and damped oscillations. A series LCR circuit gives the same equation, with the damping case (under-, critically or over-damped) deciding whether the response rings, settles quickly or crawls.
Today: the oscillation and damping idea is the basis of control systems. PID controllers, autoscalers that overshoot and settle, and the “flapping” you see when a system over-reacts to load are second-order behaviour.
Unit 3: Laplace transforms
The Laplace transform turns a function of time f(t) into a function of a complex variable s:
L{f(t)} = F(s) = ∫₀^∞ e−st f(t) dt
Its value is that it turns differential equations into algebra, which you solve and transform back.
Standard transforms
| f(t) | F(s) |
|---|---|
| 1 | 1 / s |
| tⁿ | n! / sn+1 |
| eat | 1 / (s − a) |
| sin at | a / (s² + a²) |
| cos at | s / (s² + a²) |
| sinh at, cosh at | a / (s² − a²), s / (s² − a²) |
Properties
- Linearity: L{a f + b g} = a F + b G.
- First shifting: L{eat f(t)} = F(s − a).
- Change of scale, multiplication by t, division by t, transform of derivatives and integrals: L{f′(t)} = sF(s) − f(0).
- Second shifting (unit step function): a delay of a in time multiplies the transform by e−as.
- Periodic functions have a one-period formula, and the convolution theorem says L{f ∗ g} = F(s) G(s).
- The Dirac delta models an instantaneous impulse.
Solving a differential equation
- Take the Laplace transform of both sides (initial conditions enter automatically).
- Solve the algebraic equation for Y(s).
- Break Y(s) into partial fractions.
- Take the inverse Laplace transform using the table.
Today: the Laplace transform is the language of control and signal processing, and its cousins, the Fourier and z-transforms, are behind audio, image compression and network analysis. If you have ever looked at “poles” in an alerting or control context, they are values of s.
Unit 4: Partial differentiation and its applications
For a function of several variables z = f(x, y), the partial derivative ∂z/∂x differentiates with respect to x while holding y constant.
Key results
| Idea | Statement |
|---|---|
| Total differential | dz = (∂z/∂x) dx + (∂z/∂y) dy |
| Chain rule | If z = f(x, y) and x, y depend on t: dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt) |
| Euler’s theorem | For a homogeneous function of degree n: x ∂f/∂x + y ∂f/∂y = n f |
| Jacobian | J = ∂(u, v) / ∂(x, y), the determinant of partial derivatives; it measures how a change of variables scales area |
| Taylor’s theorem (two variables) | Expands f(x, y) around a point using partial derivatives |
Maxima and minima
For f(x, y), find the stationary points by setting fx = 0 and fy = 0. Then at each point compute r = fxx, s = fxy, t = fyy:
| Condition | Result |
|---|---|
| rt − s² > 0 and r > 0 | Minimum |
| rt − s² > 0 and r < 0 | Maximum |
| rt − s² < 0 | Saddle point (neither) |
| rt − s² = 0 | Test is inconclusive |
For constrained problems, Lagrange’s method of multipliers optimises f subject to a constraint g = 0 by solving ∇f = λ∇g.
Today: this is the mathematics inside machine learning. Training a model means minimising a loss function of many variables using its partial derivatives (the gradient), and “gradient descent” walks downhill along them. The saddle-point case is the reason optimisation in high dimensions is tricky.
Unit 5: Multiple integrals
A double integral ∬ f(x, y) dA sums a function over a region of the plane. A triple integral ∭ f dV does so over a volume.
Evaluating them
- Iterated integration: integrate one variable at a time. Limits come from the region; for a rectangle they are constants, otherwise one limit depends on the other variable.
- Change of order of integration: sketch the region, then swap which variable is outer. This often turns an impossible integral into an easy one.
- Change of variables: use the Jacobian, dx dy = |J| du dv. For polar coordinates x = r cos θ, y = r sin θ, so dx dy = r dr dθ.
- Spherical and cylindrical coordinates help with triple integrals over spheres and cylinders.
Applications
| Quantity | Integral |
|---|---|
| Area | ∬ dA |
| Volume | ∬ z dA, or ∭ dV |
| Mass | ∬ ρ dA, or ∭ ρ dV (density ρ) |
| Centre of gravity | Weighted average of position using the same integrals |
| Moment of inertia | ∬ r² dm |
A classic example is the Gaussian integral ∫ e−x² dx = √π over the whole line, which is solved by converting the double integral to polar form. It is the reason the normal distribution has the constant it does.
Today: probability densities are integrated to give probabilities, and expected values are integrals. Rendering, physics simulation and image processing are numerical multiple integrals over pixels and volumes.
One-minute recap
| Unit | Key idea | You meet it in |
|---|---|---|
| 1. First-order DEs | Separate, integrating factor, exact; exponential laws | Decay of metrics, cooling, retry back-off |
| 2. Higher-order DEs | y = yc + yp; roots decide the shape | Oscillation, damping, control loops |
| 3. Laplace transforms | Differential equations become algebra | Control systems, signal processing |
| 4. Partial differentiation | Gradient, Jacobian, maxima and minima, Lagrange | Machine learning, optimisation |
| 5. Multiple integrals | Iterate, change order, change variables | Probability, simulation, rendering |
Next in Engineering Revisited: Electronic Devices & Circuits.
Written by Claude, Anthropic’s AI assistant, as part of Engineering Revisited, a quick refresher of the B.Tech subjects one at a time.