MZB127 Chap.4 First-Order Differential Equations
First-Order Differential Equations
First-order differential equations connect a state to its rate of change. The central skill is structural classification: separable equations isolate variables, linear equations use an integrating factor, and Euler's method advances numerically from local slopes. Initial conditions select a particular trajectory from a general family. Every result is checked by substitution, initial value, equilibrium and long-run behaviour.
The equation often encodes an engineering balance, so signs and units matter before any solution technique is chosen. A separable form groups all state dependence with the state differential and all independent-variable dependence on the other side. A linear form exposes a coefficient function and forcing term; its integrating factor converts the left side into one product derivative.
Some equations fit both classes and provide a valuable cross-check. Euler's method instead follows the current local slope by fixed increments, producing a table of approximate states rather than a closed formula. Equilibria arise where the rate is zero and give qualitative predictions about long-run motion.
Comparing those predictions with an exact or numerical trajectory detects wrong exponential signs, missing constants and update errors. Classification, calculation, initial data and verification are thus distinct stages of the same model solution.
What this chapter covers
- 01
Rates and engineering balance models
- 02
Separable first-order equations
- 03
Linear standard form
- 04
Integrating-factor method
- 05
Euler approximation and step size
Worked example · free
Solve a forced linear equation
- 1Identify standard linear form and the integrating factor.
- 1Convert the left side to a product derivative.
- 1Integrate and use the initial condition.
- 1Check the equation and equilibrium.
Key terms
- Separable equation
- A differential equation whose dependent-variable and independent-variable factors can be isolated on opposite sides.
- Linear equation
- A first-order equation writable as y prime plus p(t)y equals q(t).
- Integrating factor
- The exponential multiplier that makes the linear left side one product derivative.
- Initial condition
- A stated function value at one input that selects a particular solution.
- Euler method
- A numerical update using the current point and slope to step forward.
First-Order Differential Equations FAQ
Can one equation be both separable and linear?
Yes. The categories describe useful structures rather than mutually exclusive labels. When both methods apply, their valid solutions should agree after constants and initial data are handled consistently.
Why is the arbitrary constant necessary?
A first-order equation normally has a family of solution curves. The constant represents that family, and the initial condition selects the one curve required by the physical problem.
What makes an Euler answer approximate?
Euler replaces the changing trajectory by short tangent-line steps. Its result depends on step size and accumulated local error, so the update rule and chosen step must be reported with the numerical values.
What is the strongest check on an exact ODE solution?
Differentiate the candidate and substitute it into the original differential equation, then test the initial condition separately. An equilibrium or long-run comparison adds a qualitative check. Agreement among the equation, initial value and predicted behaviour is stronger than recognising a familiar exponential form by appearance alone.
Exam move
Practise classification as a separate task before solving. For integrating factors, show standard form, the factor, the product derivative and the initial condition. For separable problems, keep variables and differentials on the correct sides and include the constant. For Euler work, record each current point, slope and update.
Verify exact solutions in the original equation and compare long-run behaviour with equilibrium predictions. Start a revision session with ten unsolved equations and label each as separable, linear, both or neither without carrying out the integration. Justify each label by rewriting the equation into the defining form.
For a separable problem, show the variable grouping, integrate both sides, preserve the arbitrary constant and apply the initial value only after a family is visible. For a linear problem, write p and q, derive the integrating factor, multiply every term and explicitly identify the product derivative.
Create a verification column that differentiates the candidate, substitutes it into the original equation and checks the initial condition. Before solving a balance model, locate equilibria and predict whether nearby states move toward or away from them. For Euler practice, use a table headed current time, current state, slope and next state; advance the time and state together and avoid using the new state prematurely.
Repeat one short computation with half the step size to observe how an approximation changes without treating the finer result as automatically exact. End with a sentence connecting the formula to growth, decay or approach to equilibrium, including units where available. This routine separates method choice from algebra and gives every result several independent checks.
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