DIFFERENTIAL EQUATIONS Learning Outcomes Know that a differential

DIFFERENTIAL EQUATIONS Learning Outcomes: • Know that a differential equation (or "DE") contains derivatives or differentials. • Know the order /degree of a differential equation • Be able to find general and particular solutions of 1 st order/1 st degree differential equations

DIFFERENTIALS dx (this means "an infinitely small change in x") dθ (this means "an infinitely small change in θ") dt (this means "an infinitely small change in t")

DIFFERENTIAL EQUATIONS A differential equation (or "DE") contains derivatives or differentials. First order DE: Contains only first derivatives Second order DE: Contains second derivatives (and possibly first derivatives also) Degree: The highest power of the highest derivative which occurs in the differential equation.

DIFFERENTIAL EQUATIONS • order 2 (the highest derivative appearing is the second derivative) • degree 1 (the power of the highest derivative is 1. ) • order 1 (the highest derivative appearing is the first derivative) • degree 5 (the power of the highest derivative is 5. ) • order 2 (the highest derivative (y")4 + 2(y')7 − 5 y = 3 appearing is the second derivative) • degree 4 (the power of the highest derivative is 4. )

Example 1: Find the general solution of Our normal method: Where did the dy go from the dy/dx? Why did it seem to disappear? Could have used differentials: 1 LOOKING AT DIFFERENTIALS

Example 2: Solve θ 2 dθ = sin(t + 0. 2) dt Solve means find an equation which contains no derivatives and satisfies the given differential equation LOOKING AT DIFFERENTIALS

Example 3: a) Find the general solution for the differential equation dy + 7 x dx = 0. b) Find the particular solution given that y(0) = 3. a) b) General solution Particular solution for given initial (boundary) conditions

Example 4: Find the particular solution of y' = 5 given that when x = 0, y = 2. b) General solution Page 74 Ex 6 Particular solution for given initial (boundary) conditions

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