HSC · Mathematics Extension 1 · Calculus · also VCE Differential Equationslive from the app
Differential Equations with Exponential Growth and Decay
This type of differential equation appears frequently in natural phenomena including population dynamics, radioactive decay, compound interest, and Newton's law of cooling. The constant k determines the rate of growth or decay, while the constant A represents the initial quantity at x equals zero, making this model particularly useful for predictive analysis.
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mathematics extension 1 · calculus · differential equations with exponential growth and decaydrag it · it is yours
the one idea
why this one carries the topic.
Substitution doesn't solve the integral — it undoes a chain rule someone already applied. You're hunting the inner function whose derivative is loitering elsewhere in the integrand, so that renaming it collapses the whole thing back into a standard form you already know.
Exponential models use dy/dt = ky where k positive gives growth and k negative gives decay, leading to the solution y = Ae^(kt) that describes populations, radioactive substances, temperature change, and financial growth over time.
what examiners catch — Students often forget to use initial conditions to find the constant A, or mix up signs when k is negative in decay problems, leading to solutions that grow instead of decrease.
what you leave with
three things, not forty.
To derive the solution, separate variables to get dy/y equals k dx, then integrate both sides yielding natural log of absolute value of y equals kx plus C, which simplifies to y equals A times e to the power of kx
The half-life formula for decay problems is t sub half equals natural log of 2 divided by k, while the doubling time for growth is also natural log of 2 divided by k
Temperature problems using Newton's law of cooling follow the form dT/dt equals negative k times the quantity T minus T sub environment, requiring adjustment before separation
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this conceptdifferential equations with exponential growth and decayExponential models use dy/dt = ky where k positive gives growth and k negative gives decay, leading to the solution y = Ae^(kt) that describes populations, radioactive substances, temperature change, and financial growth over time.
sits under itfirst order differential equations and separation of variablesbecause solving dN/dt = kN is separation's first victory
sits under itdirection fields and slope fields for qualitative analysisbecause see the family before you solve for one member
the rest of calculus
7 more, same treatment.
Each one is its own knowscape in the app — built for how a particular student takes things in, not one explanation handed to everybody.
implicit differentiation and related ratesin the appintegration by substitutionin the appdifferentiating inverse circular functionsin the appdirection fields and slope fields for qualitative analysisin the appvolumes of solids of revolutionin the apparea under and between curvesin the appfirst order differential equations and separation of variablesin the app
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