MATH 395 PROBLEMS 7

IGOR KRIZ

Regular problems:

1. For the following system of linear differential equations, find an equivalent system of first order linear differential equations:

y′′ = 4x^{2}y′ + (sin(x))z′ + xy + \frac{z}{x} + ln(x)

z′′ = (tan(x))y′ + (arctan(x))z′ + \frac{y}{1+x^{2}} + x^{3}z + e^{x}.

2. Using the Wronskian, prove that the functions x + 1, e^{x} cannot be solutions of the same homogeneous linear differential equation

y′′ + a_{1}(x)y′ + a_{2}(x)y = 0

for any continuous functions

a_{1}, a_{2} : ℝ → ℝ.

3. Solve:

y′′′ − 6y′′ + 9y′ − 4y = 0.

4. Solve:

(a) y′′ − 3y′ + 2y = e^{3x},

(b) y′′ − 3y′ + 2y = x.

Challenge problem:

5. Prove that if y_{1}, …, y_{n} : ℝ → ℝ are functions with n continuous derivatives, and such that their Wronskian is everywhere non-zero, then there exists a homogenous order n linear differential equation whose space of solutions has basis y_{1}, …, y_{n}.