Lesson 9 of 9 · 11 min

Chapter review

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13 facts

1Order: the highest derivative present. Degree: its power, defined only when the equation is a polynomial in the derivatives.
2Order and degree, when defined, are positive integers.
3Clear radicals in the derivatives before reading the degree.
4A general solution of an order-n equation has n arbitrary constants; a particular solution has none.
5Separable: dy/dx = g(x)h(y) → ∫dy/h(y) = ∫g(x)dx + C.
6dP/dt = kP gives P = P₀e^(kt); doubling time (log 2)/k.
7At 10% a year compounded continuously, money doubles in 10 log 2 ≈ 6.93 years.
8Homogeneous: dy/dx = g(y/x); substitute y = vx, dy/dx = v + x dv/dx.
9F(x, y) is homogeneous of degree n if F(λx, λy) = λⁿF(x, y).
10Linear: dy/dx + Py = Q; IF = e^(∫P dx); y × IF = ∫Q × IF dx + C.
11Linear in x: dx/dy + P₁x = Q₁; IF = e^(∫P₁ dy).
12e^(log x) = x, e^(−log x) = 1/x and e^(∫tan x dx) = sec x are the integrating factors met most often.
13Use the initial condition only after the constant of integration is in place.

Common traps

Reading the degree of √(1 + (y′)²) = y″ as 1.

Square first: 1 + (y′)² = (y″)², so the degree is 2.

Giving a degree for an equation with sin(y′) or e^(y′).

Such an equation is not a polynomial in the derivatives, so its degree is not defined; the order still is.

Taking the largest power anywhere as the degree.

Only the power of the highest-order derivative counts.

Integrating to log|y| = x² with no constant, then finding that y(0) = 3 cannot be met.

Write the constant at the moment of integrating (log|y| = x² + c), then use the condition to find it.

Finding IF from dy/dx + Py = Q before the coefficient of dy/dx is 1.

Divide through by that coefficient first, then read P.

Writing the integrating factor as e^P instead of e^(∫P dx).

Integrate P first: for P = 1/x, IF = e^(log x) = x.

Forgetting to put v back as y/x at the end of a homogeneous equation.

The answer must be in x and y; substitute v = y/x in the last line.

7 to know

Separable

dy/dx = g(x)h(y) ⇒ ∫ dy/h(y) = ∫ g(x) dx + C

h(y) ≠ 0.

Exponential growth

dP/dt = kP ⇒ P = P₀e^(kt)

k < 0 gives decay.

Doubling time

T = (log 2)/k

Half-life (log 2)/λ for decay.

Homogeneous substitution

y = vx, dy/dx = v + x dv/dx

For dy/dx = g(y/x).

Integrating factor

IF = e^(∫P dx)

dy/dx + Py = Q in standard form.

Linear solution

y × IF = ∫ Q × IF dx + C

Divide by IF at the end.

Linear in x

x × e^(∫P₁ dy) = ∫ Q₁ e^(∫P₁ dy) dy + C

For dx/dy + P₁x = Q₁.

6 terms

Differential equation: An equation involving derivatives of an unknown function.
Order: The order of the highest derivative in the equation.
Degree: The power of the highest-order derivative, when the equation is a polynomial in the derivatives.
General solution: A solution with arbitrary constants, one for each order of the equation.
Particular solution: A solution with the constants fixed, usually by initial conditions.
Integrating factor: The multiplier e^(∫P dx) that turns the left side of a linear equation into an exact derivative.