Chapter 7: Differential equations Notes

Sections

1. 7A Differential equations and their solutions Mastered

   What a differential equation is, its order, checking that a function is a solution, general and particular solutions, families of solution curves, and using initial conditions to find constants.

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2. 7B Slope fields Mastered

   Drawing and reading slope fields (direction fields) for first-order differential equations, sketching solution curves through a point, matching DEs to slope fields, and spotting constant solutions.

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3. 7C Separable differential equations Mastered

   Solving separable differential equations dy/dx = f(x)g(y) by separating the variables and integrating both sides, finding constant solutions first, handling absolute values, and using initial conditions.

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4. 7D Equations of the form dy/dx = g(y) and logistic equations Mastered

   Solving dy/dx = g(y) by taking reciprocals, exponential growth, the logistic equation dP/dt = kP(1 − P/K), its solution, carrying capacity, constant solutions, and where the growth is fastest.

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5. 7E Applications of differential equations Mastered

   Modelling with differential equations, including exponential growth and decay, motion with air resistance and terminal velocity, mixing problems, the spread of a rumour or disease with the logistic model, and interpreting solutions.

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