Differential Equations · Lesson 08

Chapter practice

Bring the chapter together: say what order an equation has and whether it is linear, check a proposed solution by substituting it, fit a constant to an initial value, read a direction field, and decide whether a solution must exist and be unique. No worked example sits above the answers.

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y^{'} = f (t, y)

y (t_{0}) = y_{0}

F (t, y, y^{'}) = 0

y^{(n)}

y = C e^{k t}

Chapter navigation

Intuition

Bring the chapter together: say what order an equation has and whether it is linear, check a proposed solution by substituting it, fit a constant to an initial value, read a direction field, and decide whether a solution must exist and be unique. No worked example sits above the answers.

A driving test puts every skill into one drive, in no announced order. Deciding which rule the moment calls for is the part being tested.

Before you practise

Decide first what is asked. A proposed solution is checked by substituting it into the equation; an initial value fixes the constant; a direction field is read from the slope $f(x,y)$ at each point; and a solution through $(x_{0},y_{0})$ exists and is the only one when $f$ and $\partial f/\partial y$ are continuous near it.

y'=f(x,y),\qquad y(x_{0})=y_{0}

Reminders

The order is the highest derivative that appears. The equation is linear when the unknown and its derivatives appear only to the first power, multiplied by functions of the input.
A general solution carries as many constants as the order of the equation, and an initial value problem fixes them.
A solution curve is tangent to every slope mark of the direction field that it passes through.
Continuity of $f$ gives a solution; continuity of $\partial f/\partial y$ as well makes it the only one.

Applications