Further Integration (Edexcel A-Level Mathematics): Quizzes

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What does the general solution of a differential equation contain?

One or more arbitrary constants

What derivative is involved in a first-order differential equation?

dydx\frac{dy}{dx}

What form does a separable differential equation take?

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

After integrating both sides of a separable equation, what appears in the solution?

A constant of integration CC

What is the first step to solve dydx=xy\frac{dy}{dx} = xy?

Separate variables: 1ydy=xdx\frac{1}{y} dy = x dx

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

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

For second-order equations with constant coefficients, what method is used?

Solving the characteristic equation

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}

For d2ydx2−3dydx+2y=0\frac{d^2y}{dx^2} - 3\frac{dy}{dx} + 2y = 0 with roots m=1,2m = 1, 2, what is y(x)y(x)?

y(x)=C1ex+C2e2xy(x) = C_1 e^{x} + C_2 e^{2x}

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