MHT-CET · 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

Variables separableWrite $\dfrac{dy}{dx} = f(x)\,g(y)$ as $\dfrac{dy}{g(y)} = f(x)\,dx$ and integrate both sides; add one constantLocked
Reducing to variables separableFor $\dfrac{dy}{dx} = f(ax + by + c)$ ,
Homogeneous equationsIf $\dfrac{dy}{dx} = F\left(\dfrac{y}{x}\right)$ ,
Linear differential equationsFor $\dfrac{dy}{dx} + Py = Q$ , multiply by $\text{I.F.} = e^{\int P\,dx}$ :

Growth and decayIf $\dfrac{dx}{dt} = kx$ , then $x = x_0e^{kt}$ : 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 $\dfrac{d\theta}{dt} = -k(\theta - \theta_0)$ ;

put

v = ax + by + c

;

the equation in

v

and

x

then separates

Locked

put

y = vx

so

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

,

separate

v

and

x

,

then put back

v = \dfrac{y}{x}

Locked

then

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

Locked

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

Locked