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Math101learn.math101.caIntroduction to Differential Equations
A rigorous orientation to differential equations, solution concepts, classification, modelling, and verification.
Precise definition
A differential equation relates an unknown function to one or more derivatives. An ordinary differential equation uses derivatives with respect to one independent variable; a partial differential equation uses partial derivatives in several variables. A solution is a sufficiently differentiable function satisfying the equation at every point of a stated interval or region.
Notation and mathematical language
The order is the highest derivative present. A linear ODE has the unknown function and derivatives to first power with coefficients depending only on the independent variable. A general solution contains free constants; a particular solution fixes them, often through initial or boundary data.
Conceptual picture
A differential equation specifies local change rather than values directly. Many functions can share the same derivative rule until data select one. Exact symbolic formulas, implicit relations, qualitative phase portraits, and numerical tables are different valid representations with different information and error characteristics.
Conditions and key results
A candidate must have the required derivatives and satisfy the equation throughout its interval. Algebraic manipulations may divide by expressions that vanish and lose solutions. A model also needs units, parameter meanings, and assumptions; mathematical solvability does not validate the real-world assumptions.
A reliable strategy
- Identify independent and dependent variables, units, parameters, and any data.
- Classify order, linearity, autonomy, and whether the task asks for exact, qualitative, or numerical information.
- Choose a method whose hypotheses match the equation and retain excluded or equilibrium cases.
- Verify by substitution, apply data, state the interval, and interpret the result in context.
Fully worked example
Interpretation and application
Differential equations model motion, epidemics, circuits, fluids, finance, and feedback. Their predictions are conditional: parameter estimation, measurement error, omitted mechanisms, and model scope must be separated from errors in solving the equation itself.
Common mistakes
Verification and reasonableness
- Differentiate the candidate and substitute both sides independently.
- Check initial or boundary data and all units.
- Compare qualitative behaviour—sign, monotonicity, equilibria—with the formula or numerical result.
Practice
- What is the order of $y''+ty'=0$?
- Is $y'+t^2y=\sin t$ linear?
- Is Euler's method exact?
Answers and brief solutions
- Second order.
- Yes; $y$ and $y'$ appear linearly and coefficients depend only on $t$.
- Generally no; it produces approximations with step-dependent error.
Further deduction
A family can be checked without fully solving. For $y=C_1\cos t+C_2\sin t$, differentiating twice gives $y''=-y$, so every choice of constants solves $y''+y=0$. Conversely, an existence theorem plus linear independence shows this two-parameter family contains every solution on the interval. Verification proves membership; completeness requires the theory matching the equation's order.
Model construction begins with a balance law. If a quantity $Q$ changes through inputs and outputs, write $Q'=\sum\text{in rates}-\sum\text{out rates}$, then express rates using state variables and parameters. Checking units precedes solving. This approach produces mixing, population, and circuit equations while keeping physical assumptions visible.
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
What is the order of $y'''-2y'=e^t$?
- The derivatives present are first and third order.
- The highest is $y'''$, so the equation is third order.
End of lesson
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