Numerical solution of differential equations

Numerical solution of differential equations Quiz

Spec 6.1 · Edexcel A-Level Further Maths: Quiz

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  1. 1. Using yn+1=yn+h(2xn+yn)y_{n+1}=y_n+h(2x_n+y_n) with x0=0x_0=0, y0=1y_0=1 and h=0.5h=0.5, find y1y_1.

    • A. 11
    • B. 22
    • C. 1.51.5
    • D. 2.52.5

  2. 2. Which is the central difference approximation to dydx\frac{dy}{dx} at xnx_n?

    • A. yn+1−yn−12h\frac{y_{n+1}-y_{n-1}}{2h}
    • B. yn+1−yn−1h\frac{y_{n+1}-y_{n-1}}{h}
    • C. yn+1−yn2h\frac{y_{n+1}-y_n}{2h}
    • D. yn+1−2yn+yn−1h2\frac{y_{n+1}-2y_n+y_{n-1}}{h^2}

  3. 3. For d2ydx2=−4y\frac{d^2y}{dx^2}=-4y, h=0.1h=0.1, yn=1y_n=1 and yn−1=0.98y_{n-1}=0.98, find yn+1y_{n+1}.

    • A. 1.061.06
    • B. 1.021.02
    • C. 0.940.94
    • D. 0.980.98

  4. 4. Why does the central difference method for dydx=f(x,y)\frac{dy}{dx}=f(x,y) need y1y_1 as well as y0y_0?

    • A. It uses yn+1y_{n+1} on both sides of the formula
    • B. Each step uses yn−1y_{n-1}, so a second starting value is needed
    • C. x0x_0 is unknown
    • D. The step length must be doubled

  5. 5. What is the usual effect of halving the step length?

    • A. The estimates are always less accurate
    • B. There is no effect
    • C. The estimates are usually more accurate, but more steps are needed
    • D. The error doubles

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