MH Board · Chapter

Differential Equations

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Order and degreeThe order is the highest derivative present; the degree is its power once the equation is free of radicals and fractions in the derivativesStart
Forming a differential equationDifferentiate as many times as there are arbitrary constants, then eliminate the constants; the order equals their numberLocked
General and particular solutionsA solution satisfies the equation; the general one has as many constants as the order, and a condition like $y(0) = 1$

fixes them for a particular solution

Locked

as

\dfrac{dy}{g(y)} = f(x)\,dx

and integrate both sides; add one constant

Locked

Reducing to variables separableFor

\dfrac{dy}{dx} = f(ax + by + c)

, put

v = ax + by + c

; the equation in

v

and

x

then separatesLocked

Homogeneous equationsIf

\dfrac{dy}{dx} = F\left(\dfrac{y}{x}\right)

, put

y = vx

so

\dfrac{dy}{dx} = v + x\dfrac{dv}{dx}

, separate

v

and

x

, then put back

v = \dfrac{y}{x}

Locked

Linear differential equationsFor

\dfrac{dy}{dx} + Py = Q

, multiply by

\text{I.F.} = e^{\int P\,dx}

: then

y\cdot\text{I.F.} = \int Q\cdot\text{I.F.}\,dx + c

Locked

use two known values to find

k

,

then the time or amount asked (population, radioactive decay, interest)

Locked

Cooling, curves and other modelsNewton's law of cooling

\dfrac{d\theta}{dt} = -k(\theta - \theta_0)

; a curve from its slope condition; set up the equation, solve it, then use the given valuesLocked