Examples and practice
Differential equations
Back to lessonPractice section 1 · Questions 1–6
Integrate a logarithmic derivative directly · Apply initial data and an inverse-square gradient law · +4 more
- Integrate a logarithmic derivative directly
- Apply initial data and an inverse-square gradient law
- Integrate a polynomial and trigonometric sum term by term
- Retain the zero solution when separating an exponential equation
- Solve the introductory proportional-decay equation
- Select the initial-value branch after a separated integration
Practice section 2 · Questions 7–12
Differentiate a substitution before replacing every dependent term · Use the product rule in a homogeneous-looking substitution · +4 more
- Differentiate a substitution before replacing every dependent term
- Use the product rule in a homogeneous-looking substitution
- Reduce a trigonometric equation and apply the initial condition
- Form a net population rate with separate growth constants
- Fit Newton cooling using the temperature excess
- Fit a logistic model using both printed integration methods
Practice section 3 · Questions 13–18
Use half-life data in a signed exponential decay model · Keep both sides of a terminal velocity when separating · +4 more
- Use half-life data in a signed exponential decay model
- Keep both sides of a terminal velocity when separating
- Sketch three upper solution relations and qualify their endpoints
- Integrate, apply an initial value and sketch
- Reduce a differential equation with a substitution
- Recognise an exact product derivative
Practice section 4 · Questions 19–24
Solve a homogeneous first-order differential equation · Separate variables and integrate by parts · +4 more
Practice section 5 · Questions 25–30
Solve logistic growth and sketch its changing curvature · Separate continuous payoff time from monthly rounding · +4 more
- Solve logistic growth and sketch its changing curvature
- Separate continuous payoff time from monthly rounding
- Select the nonnegative branch of an integrated equation
- Reduce order and respect a singular boundary
- Keep the solution lost when separating a homogeneous equation
- Use two point conditions and a slowing-distance model
Practice section 6 · Questions 31–35
Derive and calibrate a logistic population model · Derive the insect model and restore its zero equilibrium · +3 more