riaology
Unit 3 · Lesson 5 of 6

An integrating factor turns a sum into a product derivative.

A system with constant input and proportional loss approaches a balance where input equals loss.

Change the input. Follow the mathematics.

What you'll be able to do
  • Set up the model and calculate its response value.
  • Connect the equation to the changing visual.
  • Explain the assumptions and identify a misleading interpretation.
Predict before exploring

What will change if you increase time? Predict the direction, then explain the response.

A model you can changeMove one control at a time.
Stored response 6.6890850294570185 units, input 2 units per minute, loss 1.3378170058914038 units per minuteIn: 2 units/minOut: 1.34Storage: 6.689 units · schematic
Time (min) versus Response y. Response, Equilibrium. Selected response value: 6.68909 units. Multiplying by the integrating factor $e^{0.2t}$ turns the left side into the derivative of a product. Equilibrium is $5b$.005101020Response yTime (min)
ResponseEquilibrium · dashed

response valuey(5)=5(2)+[1−5(2)]e−0.2(5)=6.68909y(5)=5(2)+[1-5(2)]e^{-0.2(5)}=6.68909

Multiplying by the integrating factor e0.2te^{0.2t} turns the left side into the derivative of a product. Equilibrium is 5b5b.

Assumptions, units, and limits

Constant-coefficient linear first-order equation; units are consistent with time in minutes.

Slope field for linear-ode, with current initial-value trajectory. Segment angles account for axis scale.State (same units as main plot)Time · 0 to 20 min

Each segment gives the local rate at that time and state. The red trajectory follows the chosen initial condition; Euler uses finite tangent steps.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

01

From the situation to an equation

For y′+a(t)y=by=b(t), multiply by exp(∫a dt). The product rule makes the left side an exact derivative; integrate, then impose the initial condition. Equilibria of constant-coefficient systems satisfy the algebraic rate balance.

y′+0.2y=b,y(0)=1,y=5b+(1−5b)e−0.2ty'+0.2y=b,\quad y(0)=1,\quad y=5b+(1-5b)e^{-0.2t}
02

Keep the reasoning visible

μ=e∫a(t)dt,(μy)′=μb\mu=e^{\int a(t)dt},\quad(\mu y)'=\mu b

Multiplying by the integrating factor e0.2te^{0.2t} turns the left side into the derivative of a product. Equilibrium is 5b5b.

03

What makes this model useful?

Constant-coefficient linear first-order equation; units are consistent with time in minutes.

The integrating factor multiplies every term. Forgetting to transform the right side changes the equation.

A worked example

Calculate the starting scenario

  1. Use time = 5 min; constant forcing b = 2 units/min.
  2. y(5)=5(2)+[1−5(2)]e−0.2(5)=6.68909y(5)=5(2)+[1-5(2)]e^{-0.2(5)}=6.68909
  3. Multiplying by the integrating factor e0.2te^{0.2t} turns the left side into the derivative of a product. Equilibrium is 5b5b. The checked quantity is response value.
Make the idea your own

Calculate. Compare. Explain.

Numeric checks allow 1% rounding error. Values near zero use a small absolute tolerance. Work through the setup before opening a solution.

Calculate and explain

Rebuild the starting calculation

Calculate the response value using time 5 min, constant forcing b 2 units/min. State the meaning of its sign and units.

Checked quantity: response value · units. Follow the definition for finite-cutoff and numerical-model questions.

Hint

For y′+a(t)y=b(t), multiply by exp(∫a dt). The product rule makes the left side an exact derivative; integrate, then impose the initial condition. Equilibria of constant-coefficient systems satisfy the algebraic rate balance.

Explore the visual solution
Stored response 6.6890850294570185 units, input 2 units per minute, loss 1.3378170058914038 units per minuteIn: 2 units/minOut: 1.34Storage: 6.689 units · schematic
Time (min) versus Response y. Response, Equilibrium. Selected response value: 6.68909 units. Multiplying by the integrating factor $e^{0.2t}$ turns the left side into the derivative of a product. Equilibrium is $5b$.005101020Response yTime (min)
ResponseEquilibrium · dashed

response valuey(5)=5(2)+[1−5(2)]e−0.2(5)=6.68909y(5)=5(2)+[1-5(2)]e^{-0.2(5)}=6.68909

Multiplying by the integrating factor e0.2te^{0.2t} turns the left side into the derivative of a product. Equilibrium is 5b5b.

Assumptions, units, and limits

Constant-coefficient linear first-order equation; units are consistent with time in minutes.

Slope field for linear-ode, with current initial-value trajectory. Segment angles account for axis scale.State (same units as main plot)Time · 0 to 20 min

Each segment gives the local rate at that time and state. The red trajectory follows the chosen initial condition; Euler uses finite tangent steps.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

For y′+a(t)y=by=b(t), multiply by exp(∫a dt). The product rule makes the left side an exact derivative; integrate, then impose the initial condition. Equilibria of constant-coefficient systems satisfy the algebraic rate balance.

y′+0.2y=b,y(0)=1,y=5b+(1−5b)e−0.2ty'+0.2y=b,\quad y(0)=1,\quad y=5b+(1-5b)e^{-0.2t}
Calculate and explain

Apply the idea to a new scenario

For forcing b=1b=1 unit/min, find y at t=10t=10 min with y(0)=1 and loss coefficient 0.2 min⁻¹.

Checked quantity: response value · units. Follow the definition for finite-cutoff and numerical-model questions.

Hint

For y′+a(t)y=b(t), multiply by exp(∫a dt). The product rule makes the left side an exact derivative; integrate, then impose the initial condition. Equilibria of constant-coefficient systems satisfy the algebraic rate balance.

Explore the visual solution
Stored response 4.458658867053549 units, input 1 units per minute, loss 0.8917317734107097 units per minuteIn: 1 units/minOut: 0.89Storage: 4.459 units · schematic
Time (min) versus Response y. Response, Equilibrium. Selected response value: 4.45866 units. Multiplying by the integrating factor $e^{0.2t}$ turns the left side into the derivative of a product. Equilibrium is $5b$.002.510520Response yTime (min)
ResponseEquilibrium · dashed

response valuey(10)=5(1)+[1−5(1)]e−0.2(10)=4.45866y(10)=5(1)+[1-5(1)]e^{-0.2(10)}=4.45866

Multiplying by the integrating factor e0.2te^{0.2t} turns the left side into the derivative of a product. Equilibrium is 5b5b.

Assumptions, units, and limits

Constant-coefficient linear first-order equation; units are consistent with time in minutes.

Slope field for linear-ode, with current initial-value trajectory. Segment angles account for axis scale.State (same units as main plot)Time · 0 to 20 min

Each segment gives the local rate at that time and state. The red trajectory follows the chosen initial condition; Euler uses finite tangent steps.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

For y′+a(t)y=by=b(t), multiply by exp(∫a dt). The product rule makes the left side an exact derivative; integrate, then impose the initial condition. Equilibria of constant-coefficient systems satisfy the algebraic rate balance.

y′+0.2y=b,y(0)=1,y=5b+(1−5b)e−0.2ty'+0.2y=b,\quad y(0)=1,\quad y=5b+(1-5b)e^{-0.2t}
Challenge the interpretation

Which explanation survives the model?

Choose the statement you can justify, then use the visual to explain why the others fail.

Test it with the visual
Stored response 6.6890850294570185 units, input 2 units per minute, loss 1.3378170058914038 units per minuteIn: 2 units/minOut: 1.34Storage: 6.689 units · schematic
Time (min) versus Response y. Response, Equilibrium. Selected response value: 6.68909 units. Multiplying by the integrating factor $e^{0.2t}$ turns the left side into the derivative of a product. Equilibrium is $5b$.005101020Response yTime (min)
ResponseEquilibrium · dashed

response valuey(5)=5(2)+[1−5(2)]e−0.2(5)=6.68909y(5)=5(2)+[1-5(2)]e^{-0.2(5)}=6.68909

Multiplying by the integrating factor e0.2te^{0.2t} turns the left side into the derivative of a product. Equilibrium is 5b5b.

Assumptions, units, and limits

Constant-coefficient linear first-order equation; units are consistent with time in minutes.

Slope field for linear-ode, with current initial-value trajectory. Segment angles account for axis scale.State (same units as main plot)Time · 0 to 20 min

Each segment gives the local rate at that time and state. The red trajectory follows the chosen initial condition; Euler uses finite tangent steps.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The reasoning. For y′+a(t)y=by=b(t), multiply by exp(∫a dt). The product rule makes the left side an exact derivative; integrate, then impose the initial condition. Equilibria of constant-coefficient systems satisfy the algebraic rate balance.

Your learning path

Can you explain the connection?

Mark complete when you can set up the calculation and explain what the visual is showing. Opening a page never completes it automatically.

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