Ordinary Differential Equations – Study Notes

1. What a differential equation is

A differential equation is any equation that links an unknown function to one or more of its derivatives. When the unknown depends on a single independent variable, so that only ordinary derivatives appear, it is an ordinary differential equation (ODE); if partial derivatives of a multivariable function appear, it is a partial differential equation. This chapter treats only ODEs, and mainly those of the first order and first degree.

2. Order and degree

The order is the order of the highest derivative that occurs — read it off directly. The degree is the power to which that highest-order derivative is raised, but only after the equation has been written as a polynomial in all of its derivatives.

Before reading the degree, two conditions must hold: every derivative must be free of fractional powers, and no derivative may sit inside a transcendental function such as \( \sin \), \( \log \) or an exponential. If these cannot be cleared — for example in \( \dfrac{d^{2}y}{dx^{2}} = \sin\!\left(\dfrac{dy}{dx}\right) \) — the degree is not defined.

Common trap. A fractional power on the independent variable \(x\) (such as \( x^{1/4} \)) never affects the degree; only powers of derivatives count. Likewise a square root or other function applied to a coefficient does not change the degree.

3. Forming a differential equation

To form the ODE of a family of curves, differentiate often enough to eliminate every arbitrary constant. The governing principle is:

order of the differential equation = number of independent arbitrary constants.

So a one-parameter family gives a first-order equation and a two-parameter family a second-order one. A classic catch is the family of all circles \( (x-h)^{2} + (y-k)^{2} = a^{2} \): it has three free constants \( h, k, a \), so its differential equation is of order three, not two.

4. Linear, homogeneous, general and particular

An ODE is linear if the unknown function and its derivatives appear only to the first power, are not multiplied together, and are not wrapped inside transcendental functions; otherwise it is non-linear. A linear equation whose right-hand side is zero is called homogeneous.

The general solution of an order-\(n\) equation contains \(n\) arbitrary constants. Fixing those constants with initial or boundary conditions yields a particular solution, which contains no arbitrary constants.

5. Methods for first-order, first-degree equations

• Variables separable. If the equation can be arranged as \( g(y)\,dy = f(x)\,dx \), integrate each side independently; a single constant of integration suffices.
• Reducible by substitution. An equation of the form \( \dfrac{dy}{dx} = f(ax + by + c) \) becomes separable under the substitution \( z = ax + by + c \).
• Homogeneous form. When \( \dfrac{dy}{dx} \) depends only on \( \dfrac{y}{x} \), substitute \( y = vx \) (so \( \dfrac{dy}{dx} = v + x\dfrac{dv}{dx} \)); the equation turns into a separable one in \(v\) and \(x\).
• Linear in \(y\). For \( \dfrac{dy}{dx} + Py = Q \) with \(P, Q\) functions of \(x\), multiply by the integrating factor \( \text{I.F.} = e^{\int P\,dx} \); the solution is \( y\,(\text{I.F.}) = \displaystyle\int Q\,(\text{I.F.})\,dx + C \).
• Linear in \(x\). If instead \( \dfrac{dx}{dy} + Px = Q \) with \(P, Q\) functions of \(y\), use \( \text{I.F.} = e^{\int P\,dy} \) and \( x\,(\text{I.F.}) = \displaystyle\int Q\,(\text{I.F.})\,dy + C \).

Common trap. The integrating-factor formula assumes the coefficient of \( \dfrac{dy}{dx} \) is \(1\). If it is not, divide through first, then identify \(P\).

6. Applications

Many word problems reduce to the same separable template:

• Growth. \( \dfrac{dx}{dt} = kx \;(k>0) \Rightarrow x = x_{0}\,e^{kt} \).
• Decay, evaporation, radioactivity. \( \dfrac{dA}{dt} = -kA \;(k>0) \Rightarrow A = A_{0}\,e^{-kt} \). The half-life is the time for \(A\) to fall to \( \tfrac{1}{2}A_{0} \).
• Newton's law of cooling. \( \dfrac{dT}{dt} = k\,(T - T_{m}) \), with \( T_{m} \) the surrounding temperature and \( k < 0 \) so the body approaches \( T_{m} \).
• Mixture in a tank. \( \dfrac{dx}{dt} = (\text{rate of salt in}) - (\text{rate of salt out}) \), where the outflow uses the current concentration \( \dfrac{x}{\text{volume}} \).

Common trap. For any decreasing quantity the proportionality constant carries a minus sign; writing \( +k \) by mistake turns decay into growth.