Advanced Calculus: Problem sheet 2

Solutions should be submitted to tutors by 1pm on Monday 19 February 2001

1. Find the general solutions of the following differential equations.

(a) y′′ + 5y′ + 6y = 0;

(b) y′′ = 0;

(c) y′′′ − 6y′′ + 11y′ − 6y = 0;

(d) y′′ + 8y′ + 25y = 0;

(e) y^{(4)} + 3y′′ − 4y = 0;

(f) y′′ − 10y′ + 35y = 0;

(g) y^{(4)} − 4y′′′ + 6y′′ − 4y′ + y = 0.

(h) y′′ + y′ + y = 0;

(i) y′′ − 7y′ + 10y = x;

(j) y′′ + 2y′ − 3y = 3x^{3};

(k) y′′ − 3y′ + 2y = 4e^{3x};

(l) y′′ − 4y′ + 4y = 10e^{2x};

(m) y′′ + 2y′ + 2y = 8x^{2} + 12e^{−x};

(n) y′′ − y′ − 2y = cos x;

(o) y′′ − 2y′ + 5y = e^{x} cos x;

(p) y′′ + 4y = cos 2x − 2 sin 2x;

(q) y′′ − 2y = 4x^{2}e^{2x};

(r) y′′ − a^{2}y = e^{bx}, where a ≠ 0, b ≠ 0.

2. Obtain the general solution of each of the following systems of simultaneous equations:

(a) y′ = y − z, z′ = z − 4y;

(b) y′ = −3y − 4z + x, z′ = y + 2z + 1;

(c) y′′ + 2z′ + y = sin x, z′′ + 2y′ + z = 0.

3. In each case you are given a second-order linear differential equation and a particular solution. Find the general solution in each case:

(a) xy′′ + (2 − x)y′ − y = 0, y(x) = 1/x;

(b) xy′′ − (2x + 2)y′ + (x + 2)y = 0, y(x) = e^{x};

(c) (1 − x^{2})y′′ − 2xy′ + 2y = 0, y(x) = x.

4. Find the solutions of the following differential equations satisfying the given conditions:

(a) y′′ − 2y′ + y = e^{x}, y(0) = 0, y′(0) = 1;

(b) y′′ − 2y′ − 2y = 6x^{2} − 16x + 10, y(0) = 1, y′(0) = −3;

(c) y′′′ − 2y′′ + y′ − 2y = 12 cos 2x + 4x, y(0) = 0, y′(0) = −2, y′′(0) = 1.

RJC 1/2/2001