A tank’s amount changes at a prescribed rate, but knowing the rate does not tell you the starting amount. Suppose the rate is 2t units per minute at time t minutes. Starting from five units, sketch a curve whose slope starts at zero and increases. Now sketch another curve with the same slopes but starting from one unit. The curves should differ by a vertical translation, not by their shape.
This is a mathematical teaching model. The numbers do not describe a measured tank. Our first task is to distinguish a rate rule, a family of functions satisfying it, and one trajectory selected by additional information.
Predict
Which statement needs checking separately: “the derivative is 2t” or “the initial amount is five”? Predict whether y=t²+1 solves the equation y′=2t, and whether it solves the initial-value problem with y(0)=5. Write two separate yes/no answers before calculating. A formula can satisfy the differential equation perfectly and still fail the requested initial condition.
Experiment
Integrating the rate gives y=t²+C, where C is an arbitrary constant. The initial condition fixes C=5. Inspect the following worked trace, and add the row for t=3 yourself.
Time t
Predicted amount t²+5
Required rate 2t
Derivative of prediction
0
5
0
0
1
6
2
2
2
9
4
4
The final column is obtained symbolically, not by subtracting adjacent amounts. Between times one and two the average rate is three, whereas the instantaneous rate at two is four. Explain why those results are compatible. A numerical table can suggest agreement; symbolic substitution checks the equation throughout an interval.
Understand
A differential equation relates an unknown function to its derivatives. The prime in y′ means differentiation with respect to t. Its order is the highest derivative that occurs: y″+t y′+y=0 has order two. This equation is linear because y and its derivatives appear to the first power with coefficients depending only on t. The equation y′=y² is nonlinear.
A solution includes both a function and an interval where the function has the necessary derivatives and satisfies the equation. For y′=y², the candidate y=1/(1−t) has derivative 1/(1−t)², equal to its square. It also has y(0)=1. Nevertheless it is undefined at t=1. The largest interval containing zero on which this formula solves the initial-value problem is the interval extending from negative infinity to one, excluding one. The separate branch beyond one is not a continuation through the singularity.
Verification does not require discovering the formula. Differentiate the candidate, substitute into both sides, check the initial data, and state a valid domain. These four checks often detect errors that a plausible graph conceals.
Apply
For temperature T approaching a fixed ambient temperature A, consider T′=−k(T−A), with k positive. If time is minutes, k has units of inverse minutes. Check the sign above and below ambient: both signs should move the temperature toward A. This model assumes proportional response and constant ambient conditions. An exact solution to the model is not evidence that those assumptions fit an actual object.
Create your own candidate solution, differentiate it, and design an equation it satisfies. Then choose an incompatible initial value and explain exactly which verification step fails. Complete the practice with units and intervals alongside the arithmetic.
Continue
Next, read an equation geometrically through its slope field before attempting a formula. Keep a two-column record of equation verification and initial-condition verification: neither substitutes for the other, and both will matter in every later method.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
For y′=3y−2t, find the required slope at (t,y)=(2,4).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For y′=3t² with y(0)=4, find y(2).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
The equation y″+t y′+y=0 has which order?
Verify y=t²+1 solves y′=2t with y(0)=1, and explain what fails if y=t²+3 is proposed for that same initial-value problem.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A synthetic temperature T approaches a fixed ambient value 20 with rate proportional to the temperature difference. Formulate a cooling equation with positive constant k and state units if time is minutes.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Verify y=1/(1−t) solves y′=y² with y(0)=1, and identify the largest interval containing zero where this formula solves the IVP.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
