I4-I6 Second order linear equations Quiz: AQA Further Maths, Unit 2

The 20 questions

1. What is the auxiliary equation of y'' + a y' + b y = 0?

   • m^2 + a + b = 0
   • m^2 - a m + b = 0
   • m^2 + a m + b = 0
   • m^2 + a m - b = 0

2. If the discriminant a^2 - 4b of the auxiliary equation is positive, what are the roots?

   • A repeated real root
   • Two distinct real roots
   • Two complex roots
   • No real roots

3. If the discriminant of the auxiliary equation is zero, what is the form of the general solution?

   • (A + Bx) e^(mx)
   • A e^(m1 x) + B e^(m2 x)
   • e^(px)(A cos(qx) + B sin(qx))
   • A + Bx

4. If the auxiliary equation has complex roots p +/- qi, what is the general solution?

   • e^(px)(A cos(qx) + B sin(qx))
   • A e^(px) + B e^(-px)
   • (A + Bx) e^(px)
   • e^(qx)(A cos(px) + B sin(px))

5. What is the complementary function of a second order linear differential equation?

   • The sum of all particular integrals
   • Any single solution of the full equation
   • The general solution of the homogeneous equation with the right-hand side set to zero
   • The particular integral alone

6. The general solution of y'' + a y' + b y = f(x) is:

   • Complementary function plus a particular integral
   • Particular integral only
   • Complementary function times particular integral
   • Complementary function minus particular integral

7. Find the general solution of y'' - 3y' + 2y = 0.

   • y = (A + Bx) e^x
   • y = A cos x + B sin x
   • y = A e^(-x) + B e^(-2x)
   • y = A e^x + B e^(2x)

8. Find the general solution of y'' + 4y' + 4y = 0.

   • y = e^(-2x)(A cos(2x) + B sin(2x))
   • y = A e^(-2x) + B e^(2x)
   • y = (A + Bx) e^(2x)
   • y = (A + Bx) e^(-2x)

9. Find the general solution of y'' + 2y' + 5y = 0.

   • y = e^(-x)(A cos(2x) + B sin(2x))
   • y = e^(x)(A cos(2x) + B sin(2x))
   • y = e^(-2x)(A cos(x) + B sin(x))
   • y = A e^(-x) + B e^(-5x)

10. A particular integral of y'' - y = 2x is which function?

    • y = 2x
    • y = -2x
    • y = -2
    • y = x^2

11. A particular integral of y'' + y = e^x is which function?

    • y = 2 e^x
    • y = e^x / 3
    • y = e^x
    • y = (1/2) e^x

12. Solve y'' + 2y' + y = 0 with y(0) = 1 and y'(0) = 0.

    • y = (1 + 2x) e^(-x)
    • y = e^(-x)
    • y = (1 - x) e^(-x)
    • y = (1 + x) e^(-x)

13. A particular integral of y'' - 2y' = 4 is which function?

    • y = -4x
    • y = -2
    • y = 2x
    • y = -2x

14. Find the particular integral of y'' - 4y' + 4y = 8.

    • y = 2x
    • y = 8
    • y = -2
    • y = 2

15. What is the auxiliary equation of y'' + 6y' + 9y = 0, and what type of root does it have?

    • A repeated root m = -3
    • Two roots m = -6 and m = -9
    • Complex roots m = -3 +/- 2i
    • Two distinct roots m = 3 and m = -3

16. Find the general solution of y'' - 2y' + 5y = 0.

    • y = A e^x + B e^(5x)
    • y = e^x(A cos(x) + B sin(x))
    • y = e^(2x)(A cos(x) + B sin(x))
    • y = e^x(A cos(2x) + B sin(2x))

17. Find a particular integral of y'' + 3y' + 2y = e^(-x).

    • x e^(-x)
    • -x e^(-x)
    • e^(-x)
    • 2x e^(-x)

18. Solve y'' - 5y' + 6y = 0 with y(0) = 2 and y'(0) = 5.

    • y = e^(2x) + 2 e^(3x)
    • y = e^(-2x) + e^(-3x)
    • y = 2 e^(2x) + e^(3x)
    • y = e^(2x) + e^(3x)

19. Find a particular integral of y'' + y' - 2y = x.

    • y = x/2 + 1/4
    • y = -x/2 - 1/4
    • y = -x/2 + 1/4
    • y = -x/2

20. Which second order equation has general solution y = A e^(2x) + B x e^(2x)?

    • y'' - 4y = 0
    • y'' - 4y' + 4y = 0
    • y'' + 4y' + 4y = 0
    • y'' - 2y' + 4y = 0

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