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MH HSC 12 Maths

Differential Equations

Each concept is a short lesson, then questions until you get three right.

0/9

9 concepts, a short lesson each

  • Describing equations0/3
  • Solving0/4
  • Applications0/2

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Order and degree

The order is the highest derivative present; the degree is its power once the equation is free of radicals and fractions in the derivatives

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Describing equations0 of 3

  1. 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
  2. Forming a differential equationDifferentiate as many times as there are arbitrary constants, then eliminate the constants; the order equals their numberStart
  3. General and particular solutionsA solution satisfies the equation; the general one has as many constants as the order, and a condition like y(0)=1y(0) = 1 fixes them for a particular solutionStart

Solving0 of 4

  1. Variables separableWrite dydx=f(x) g(y)\dfrac{dy}{dx} = f(x)\,g(y) as dyg(y)=f(x) dx\dfrac{dy}{g(y)} = f(x)\,dx and integrate both sides; add one constantStart
  2. Reducing to variables separableFor dydx=f(ax+by+c)\dfrac{dy}{dx} = f(ax + by + c), put v=ax+by+cv = ax + by + c; the equation in vv and xx then separatesStart
  3. Homogeneous equationsIf dydx=F(yx)\dfrac{dy}{dx} = F\left(\dfrac{y}{x}\right), put y=vxy = vx so dydx=v+xdvdx\dfrac{dy}{dx} = v + x\dfrac{dv}{dx}, separate vv and xx, then put back v=yxv = \dfrac{y}{x}Start
  4. Linear differential equationsFor dydx+Py=Q\dfrac{dy}{dx} + Py = Q, multiply by I.F.=e∫P dx\text{I.F.} = e^{\int P\,dx}: then y⋅I.F.=∫Q⋅I.F. dx+cy\cdot\text{I.F.} = \int Q\cdot\text{I.F.}\,dx + cStart

Applications0 of 2

  1. Growth and decayIf dxdt=kx\dfrac{dx}{dt} = kx, then x=x0ektx = x_0e^{kt}: use two known values to find kk, then the time or amount asked (population, radioactive decay, interest)Start
  2. Cooling, curves and other modelsNewton's law of cooling dθdt=−k(θ−θ0)\dfrac{d\theta}{dt} = -k(\theta - \theta_0); a curve from its slope condition; set up the equation, solve it, then use the given valuesStart

Practise Differential Equations

Once the concepts click, try the questions your exam asks.

Textbook exercisesEvery exercise, with model answersPast board questionsAsked from this chapter, with solutions