Question

What is the general solution of the DE with n linearly independent solutions u1(t), …., un(t) of a nth order linear homogeneous DE?

(a) \(u(t)=u_1 (t)+⋯+c_{n+1} u_n (t)=∑_{k+1}^n=c_{k+1} u_k (t)\)

(b) \(u(t)=u_1 (t)+⋯+u_n (t)=∑_{k=1}^nu_k(t) \)

(c) \(u(t)=c_1 u_1 (t)+⋯+c_n u_n (t)=∑_{k=1}^n c_k u_k (t) \)

(d) \(u(t)=c_0 u_0 (t)+⋯+c_n u_n (t)=∑_{k=0}^∞c_k u_k (t) \)

This question was addressed to me in a job interview.

The origin of the question is Homogeneous Linear PDE with Constant Coefficient topic in portion Partial Differential Equations of Engineering Mathematics

Answer 1

Correct choice is (c) \(u(t)=c_1 u_1 (t)+⋯+c_n u_n (t)=∑_{k=1}^n c_k u_k (t) \)

The best I can explain: If we know n linearly independent solutions u1(t), …., un(t) of a nth order linear homogeneous DE, then the general solution of this DE has the form:

\(u(t)=c_1 u_1 (t)+⋯+c_n u_n (t)=∑_{k=1}^nc_k u_k (t)\)