MATH 395 PROBLEMS 8

IGOR KRIZ

Regular problems:

1. Find all solutions of the linear differential equation y′′ − 6y′ + 8y = x^{3}.

2. Find a basis of the space of solutions of y′′′ = y.

3. Find a basis of the space of solutions of y′_{1} = 3y_{1} − 2y_{3}

y′_{2} = 2y_{1} + y_{2} − 2y_{3}
y′_{3} = y_{1}

4. Find all solutions of

\begin{matrix}y′_{1} = y_{1} − 2y_{2} + e^{3x} \\ y′_{2} = −2y_{1} + y_{2} + e^{4x} \end{matrix}

Challenge problem:

5. Find all possible Wronskians of n-tuples of solutions of

(a) The homogeneous system (L):

y′_{i} = ∑_{j=1}^{n} a_{ij}(x)y_{i}, i = 1, …, n

(b) The homogeneous system (L̃):

y^{(n)} + a_{1}(x)y^{(n−1)} + ⋯ + a_{n}(x)y = 0.

[The Wronskian satisfies a first order differential equation. Find the equation.]