General Solutions (AQA A-Level Mathematics): Quizzes

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What does the general solution of a differential equation typically contain?

One or more arbitrary constants

Which derivative appears in a first-order differential equation?

The first derivative dydx\frac{dy}{dx}

What form can a separable differential equation be written in?

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

What is CC in the expression ln⁡∣y∣=x22+C\ln |y| = \frac{x^2}{2} + C?

The constant of integration

What is the general solution of dydx=xy\frac{dy}{dx} = xy?

y=C1ex22y = C_1 e^{\frac{x^2}{2}}

For separable equations, what is integrated?

∫1h(y)dy=∫g(x)dx\int \frac{1}{h(y)} dy = \int g(x) dx

What is the characteristic equation for d2ydx2−3dydx+2y=0\frac{d^2y}{dx^2} - 3\frac{dy}{dx} + 2y = 0?

m2−3m+2=0m^2 - 3m + 2 = 0

For distinct roots m1m_1 and m2m_2, what is the general solution form?

y(x)=C1em1x+C2em2xy(x) = C_1 e^{m_1 x} + C_2 e^{m_2 x}

What are the roots of the characteristic equation m2−3m+2=0m^2 - 3m + 2 = 0?

m=1m = 1 and m=2m = 2

How is a second-order equation with constant coefficients solved?

By solving the characteristic equation

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