Differential Equations (AQA A-Level Mathematics): Quizzes

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Solving & Interpreting Differential Equations
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What is the standard form of a separable differential equation?

dydx=g(x)h(y)\frac{dy}{dx} = g(x)h(y)

In solving separable equations, where should all terms involving yy be moved?

To one side

What is the integrating factor for dydx+P(x)y=Q(x)\frac{dy}{dx} + P(x)y = Q(x)?

e∫P(x) dxe^{\int P(x) \, dx}

What form defines a linear first-order differential equation?

dydx+P(x)y=Q(x)\frac{dy}{dx} + P(x)y = Q(x)

Why multiply by the integrating factor in linear first-order equations?

Makes left side an exact derivative

What is the standard form of a second-order homogeneous differential equation?

d2ydx2+pdydx+qy=0\frac{d^2y}{dx^2} + p\frac{dy}{dx} + qy = 0

What substitution creates the characteristic equation for second-order equations?

y=emxy = e^{mx}

What is the purpose of initial conditions in differential equations?

Find specific values of constants

In y(t)=P0ekty(t) = P_0e^{kt} for population growth, what does P0P_0 represent?

Initial population

For y(t)=e−kt(Acos⁡(ωt)+Bsin⁡(ωt))y(t) = e^{-kt}(A\cos(\omega t) + B\sin(\omega t)) with k>0k > 0, what happens as tt increases?

Oscillations dampen over time

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