Differential equations: models, methods and verification
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 240 minutes, as an unvalidated planning estimate. This collection contains 12 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Choose and justify a method before opening the solution. Include the assumptions, intermediate work and interpretation that make the answer defensible.
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Let q(t)=r(t−1)−r(t−2)−H(t−3), where r(t)=max(t,0). A state starts at rest and satisfies Dx=B(b)x+e₁ Dq, with B(b)=[−2,1;b,−2] and e₁=(1,0). (1) Derive Dq and D²q from jumps and slopes. (2) Classify the homogeneous origin for every real b, including boundaries. (3) At b=0 construct normalized E(t), derive the causal forced response and the jump at3. (4) With x=S z, S=[1,0;1,1], construct the changed generator and flow at b=0 and the transformed state jump. Explain why an impulse does not automatically make a stable homogeneous system unstable.
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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 changed oscillator obeys y″+2y′+9y=cos2t+(1/3)cos3t. Derive its transfer function and attracting periodic response. Compare the frequency maximizing displacement under unit sinusoidal force with the one maximizing velocity. For initial state y(0)=1,y′(0)=0, specify the free correction. Finally remove damping: which forcing term now prevents a bounded periodic solution, and what secular term is required? Explain why a triangle-forcing tail certificate from the lesson cannot simply certify this changed 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.
Solve y′=y²,y(0)=2, verify it and give its maximal interval containing zero. State an equilibrium omitted if separation divides by y². Explain why local uniqueness does not imply a solution for every real time.
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.
For y′=3y(1−y/12), classify equilibria and signs for nonnegative populations. Solve y(0)=3 and find the time at which y=6. Compare initial growth with the exponential model y′=3y having the same initial value; explain the capacity assumption.
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 perfectly mixed tank initially holds 40 L with 10 g solute. Inflow is 2 L/min at 3 g/L and outflow is 2 L/min. Write and solve the amount equation; give concentration and its limit. Show the integrating-factor step and units.
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.
For y′=−5y,y(0)=1, write the Euler value after n steps of width h and the exact value at the same time. Compare h=0.5 and h=0.1, including stability, sign and a concrete two-step error. Explain why a smaller stable step is not automatically exact.
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.
Give real general solutions of y″−y′−6y=0, y″+6y′+9y=0, and y″+2y′+5y=0. For the repeated-root equation impose y(0)=2,y′(0)=1. Explain why two copies of the same exponential cannot fit arbitrary two initial conditions.
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.
Solve y″+4y=cos2t with zero initial displacement and velocity. Explain resonance. Then for y″+2y′+4y=0 use E=(y′)²/2+2y² to show what damping does, and classify its roots.
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.
For y″+y=f(t), y(0)=y′(0)=0, derive an integral solution using the fundamental pair cos t,sin t. Evaluate it for f(t)=2 on t≥0 and verify. Explain the causal upper limit and the role of the Wronskian.
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.
Solve x′=−2x+y, y′=−2y with x(0)=0,y(0)=3 and classify the origin. Explain why eigenvalues alone do not justify diagonalization. Separately compare the origins of x′=−x³,y′=−y and x′=x³,y′=−y: their Jacobians agree there, so what can linearization conclude and what does the nonlinear equation show?
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.
Use a one-sided Laplace transform to solve y′+3y=6,y(0)=2, and then the same equation with y(0)=0. Identify the zero-input and zero-state parts for general y(0)=a. State a valid transform convergence half-plane for the resulting bounded solutions.
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.
Solve y′+2y=u(t−1)+3δ(t−2), with zero prehistory and y(0)=0, using causal notation. State continuity or jump behavior at t=1 and t=2, and verify the impulse jump by integration. Explain why u(t−1)e^(−2t) is not the same shifted response as u(t−1)e^(−2(t−1)).
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.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
Let q(t)=r(t−1)−r(t−2)−H(t−3), where r(t)=max(t,0). A state starts at rest and satisfies Dx=B(b)x+e₁ Dq, with B(b)=[−2,1;b,−2] and e₁=(1,0). (1) Derive Dq and D²q from jumps and slopes. (2) Classify the homogeneous origin for every real b, including boundaries. (3) At b=0 construct normalized E(t), derive the causal forced response and the jump at3. (4) With x=S z, S=[1,0;1,1], construct the changed generator and flow at b=0 and the transformed state jump. Explain why an impulse does not automatically make a stable homogeneous system unstable.
A changed oscillator obeys y″+2y′+9y=cos2t+(1/3)cos3t. Derive its transfer function and attracting periodic response. Compare the frequency maximizing displacement under unit sinusoidal force with the one maximizing velocity. For initial state y(0)=1,y′(0)=0, specify the free correction. Finally remove damping: which forcing term now prevents a bounded periodic solution, and what secular term is required? Explain why a triangle-forcing tail certificate from the lesson cannot simply certify this changed problem.
Solve y′=y²,y(0)=2, verify it and give its maximal interval containing zero. State an equilibrium omitted if separation divides by y². Explain why local uniqueness does not imply a solution for every real time.
For y′=3y(1−y/12), classify equilibria and signs for nonnegative populations. Solve y(0)=3 and find the time at which y=6. Compare initial growth with the exponential model y′=3y having the same initial value; explain the capacity assumption.
A perfectly mixed tank initially holds 40 L with 10 g solute. Inflow is 2 L/min at 3 g/L and outflow is 2 L/min. Write and solve the amount equation; give concentration and its limit. Show the integrating-factor step and units.
For y′=−5y,y(0)=1, write the Euler value after n steps of width h and the exact value at the same time. Compare h=0.5 and h=0.1, including stability, sign and a concrete two-step error. Explain why a smaller stable step is not automatically exact.
Give real general solutions of y″−y′−6y=0, y″+6y′+9y=0, and y″+2y′+5y=0. For the repeated-root equation impose y(0)=2,y′(0)=1. Explain why two copies of the same exponential cannot fit arbitrary two initial conditions.
Solve y″+4y=cos2t with zero initial displacement and velocity. Explain resonance. Then for y″+2y′+4y=0 use E=(y′)²/2+2y² to show what damping does, and classify its roots.
For y″+y=f(t), y(0)=y′(0)=0, derive an integral solution using the fundamental pair cos t,sin t. Evaluate it for f(t)=2 on t≥0 and verify. Explain the causal upper limit and the role of the Wronskian.
Solve x′=−2x+y, y′=−2y with x(0)=0,y(0)=3 and classify the origin. Explain why eigenvalues alone do not justify diagonalization. Separately compare the origins of x′=−x³,y′=−y and x′=x³,y′=−y: their Jacobians agree there, so what can linearization conclude and what does the nonlinear equation show?
Use a one-sided Laplace transform to solve y′+3y=6,y(0)=2, and then the same equation with y(0)=0. Identify the zero-input and zero-state parts for general y(0)=a. State a valid transform convergence half-plane for the resulting bounded solutions.
Solve y′+2y=u(t−1)+3δ(t−2), with zero prehistory and y(0)=0, using causal notation. State continuity or jump behavior at t=1 and t=2, and verify the impulse jump by integration. Explain why u(t−1)e^(−2t) is not the same shifted response as u(t−1)e^(−2(t−1)).
