Differential Equations: Engineering Checkpoint

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The idea

A useful differential-equation solution starts with a model, not a favorite formula. Identify the state, independent variable, input and initial or boundary data. Then decide whether the equation is separable, linear, a constant system, a transform problem or primarily numerical.

Visual guide

VISUAL GUIDEA response must match both the forcing and the initial state

State y = 2 − e⁻²ᵗ
Rate y′ = 2e⁻²ᵗ
Equilibrium y = 2

For y′ + 2y = 4 with y(0) = 1, the solution 2 − e⁻²ᵗ starts at 1 and approaches 2. Its derivative is positive and tends to zero, agreeing with the rate law 4 − 2y throughout the motion.

Method and assumptions

After solving, check the original equation and the data. Inspect units, equilibria, signs and long-time behavior. If using a numerical method, compare at a fixed endpoint under step refinement and check its stability restriction. Qualitative agreement supports a calculation but cannot replace an error estimate.

Worked example: a switched circuit

For $v'+v=H(t-2)$ with $v(0)=0$ , $v=0$ before the switch and $v=1-e^{-(t-2)}$ afterward. Voltage stays continuous at $t=2$ , while its derivative changes from $0$ to $1$ . A step input is not an impulse, so a voltage jump would contradict the equation.

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

A computed response fits the ODE but misses the required initial value. Is it a solution of that initial-value problem?

Yes; the ODE is enough.

Yes; initial values are optional.

No; both the equation and initial value must hold.

Hint 1 · Find a starting point

Separate checking the differential equation from checking the supplied data.

Hint 2 · Take the next step

A solution family can contain many curves, but the initial value selects one.

Show the reasoning

Answer: No; both the equation and initial value must hold.

Verification must include the model, initial conditions and the interval on which the response is valid.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

Worked example: a stiff approximation

For $y'=-100y$ , $y(0)=1$ , exact decay is $e^{-100t}$ . Euler with $h=0.03$ has factor $-2$ and diverges. A smaller step must satisfy $h<0.02$ for stability; accuracy may require a much smaller value. Backward Euler is stable for positive steps but still has discretization error.

Interpreting the result

The expanded course supplies a foundation, not every special-function or partial-differential-equation method. Use the checkpoint to identify gaps: a wrong initial slope suggests missing homogeneous terms; an unexpected jump suggests a forcing error; alternating growth in a decaying model suggests numerical instability.

Practice

Solve $y'+2y=4$ , $y(0)=1$ .
Find the zero-state response to $x''+x=\cos t$ .
Classify a linear system with eigenvalues $-2\pm3i$ .

Show worked solutions

$y=2-e^{-2t}$ ; substitution and $y(0)=1$ both check.
$x=(t/2)\sin t$ . Resonance gives an increasing envelope, and both initial values are zero.
A stable spiral: radius-like amplitudes decay exponentially while the state rotates.

Further study

MIT OpenCourseWare: Differential Equations provides a full university course with additional lectures and exercises.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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