PHYS2150 Chap.7 First-Order Differential Equations

First-Order Differential Equations

Classification controls the solution path

A first-order differential equation links an unknown function with its first derivative, but similar-looking equations can require different methods. Begin by arranging the equation and naming the structure that is actually present.

Separable equations allow all factors involving the dependent variable to accompany its differential and all factors involving the independent variable to move to the other side. Linear equations take the normalized form y′+P(x)y=Q(x). Homogeneous first-order equations can be expressed through the ratio y/x and invite y=vx.

Exact equations have the differential form M dx+N dy=0 and pass the cross-partial test M_y=N_x on the region of interest. This classification line is part of the solution because it justifies the transformation that follows.

Separation needs a lost-solution check

Separating variables often requires division by a function of y.

Before dividing, set that factor equal to zero and test the resulting constant functions in the original equation. These equilibrium solutions may be the most important states in a physical model, and division can remove them silently. For an autonomous equation y′=f(y), roots of f divide the phase line into intervals.

The sign of f on each interval indicates whether solutions rise or fall and whether an equilibrium attracts or repels nearby states. Integrate the nonconstant branches only after this qualitative map is recorded.

Logarithms, denominators and square roots in the resulting relation can restrict the interval, while the initial value determines which connected branch is relevant.

Integrating factors manufacture a derivative

For a linear equation, first divide by the coefficient of y′ so that its coefficient is one on an interval where that division is valid.

The integrating factor μ=exp(∫P dx) is designed so that multiplying every term turns μy′+μPy into (μy)′. Display that product derivative rather than treating μ as a memorized decoration. Integration then produces μy as an antiderivative of μQ plus a constant. Solve for y, apply the initial condition, and state an interval on which P, Q and the solution are defined.

The final check has two independent parts: substitution into the differential equation and evaluation of the initial data.

Substitution and exactness recover hidden structure

When dependence occurs through y/x, the substitution y=vx gives y′=v+xv′ and commonly reduces the equation to a separable one in v and x. Back-substitute fully and check whether any branch was excluded.

For an exact equation, seek a potential F with F_x=M and F_y=N. Integrate one component while retaining an unknown function of the other variable, then differentiate and compare with the remaining component to determine that function. The implicit relation F=C is often the cleanest final form.

If exactness fails, a single-variable integrating factor may sometimes repair the equation when the standard diagnostic ratio depends on only that variable.

Physical models require interpretation

Rate models begin with a balance: accumulation equals input minus output. Every term must share the derivative’s units.

Exponential growth and decay place an inverse-time parameter beside the stored quantity; cooling uses a difference from ambient temperature; mixing models distinguish solute amount from concentration and account for changing volume. After solving, compare signs, equilibria and long-time behaviour with the verbal system.

A positive decay constant should not create growth, and an undriven cooling model should approach its ambient equilibrium. Define variables and units, impose the data, state the physical interval, and interpret the limiting branch. These checks turn a symbolic family into a defensible model rather than a detached antiderivative.

What this chapter covers

• 01

  Equation classification

• 02

  Separable equations and equilibria

• 03

  Linear integrating factors

• 04

  Homogeneous substitutions

• 05

  Exact equations and potentials

Worked example · free

Solve a linear initial-value problem

Key terms

Equilibrium Solution

A constant solution obtained from a zero of the rate function, so the derivative vanishes for every value of the independent variable; its stability is read from nearby derivative signs.

Initial Value Problem

A differential equation paired with the value of the unknown function, and possibly derivatives for higher-order problems, at one starting point to select a particular solution curve.

Integrating Factor

A nonzero multiplier chosen so a differential equation acquires an exact or product-derivative form; for a normalized first-order linear equation it is exp(∫P dx).

Exact Equation

An equation M dx+N dy=0 whose coefficients are partial derivatives of one potential function, typically verified locally by the equality M_y=N_x.

Phase Line

A one-dimensional sign diagram for an autonomous equation that marks equilibria and shows whether solutions increase or decrease between them.

First-Order Differential Equations FAQ

Exam move

Build a one-page method table with columns for diagnostic form, legal transformation, solutions that might be lost, and final verification. During practice, spend the first line classifying before doing algebra. For separable and autonomous equations, list equilibria and sketch a phase line. For linear equations, normalize first and explicitly show the product derivative.

For homogeneous substitutions, write y′=v+xv′ before replacement. For exact equations, perform the cross-partial test and retain the missing single-variable function. Finish every solution with three separate audits: substitute into the original equation, apply the data, and state the valid physical branch with units and limiting behaviour.

Working through First-Order Differential Equations in PHYS2150? Sia is AskSia’s AI Physics tutor — ask any PHYS2150 First-Order Differential Equations question and get a clear, step-by-step explanation grounded in how PHYS2150 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.