MATH528

In this mini-lab, we'll investigate the method of variation of constants for solving inhomogeneous ODEs of the form (§2.10)

One of the objectives of this mini-lab is to show how combined active-reading, hand-computation, structured writing, scratch notes, and symbolic computation come together to enable deeper understanding of topics within mathematical physics.

1Method of variation of parameters

Seek solution of (1) as

y (x) = y_{h} (x) + y_{p} (x)

, with

y_{h} (x)

the general solution of the homogeneous ODE

In[22]:=

rhs[y_,x_,p_,q_] := D[y,{x,2}] + p[x] D[y,x] + q[x] y;

rhsODE = rhs[u[x] Subscript[y,1][x] + v[x] Subscript[y,2][x],x,p,q]

$p (x) (y_{1} (x) u^{'} (x) + u (x) y_{1}^{'} (x) + y_{2} (x) v^{'} (x) + v (x) y_{2}^{'} (x)) + q (x) (u (x) y_{1} (x) + v (x) y_{2} (x)) + y_{1} (x) u^{''} (x) + 2 u^{'} (x) y_{1}^{'} (x) + u (x) y_{1}^{''} (x) + y_{2} (x) v^{''} (x) + 2 v^{'} (x) y_{2}^{'} (x) + v (x) y_{2}^{''} (x)$

In[24]:=

Simplify[rhsODE /. {Subscript[y,1]”[x] -> - p[x] Subscript[y,1]'[x] - q[x] Subscript[y,1][x],Subscript[y,2]”[x] -> - p[x] Subscript[y,2]'[x] - q[x] Subscript[y,2][x]}]

$p (x) (y_{1} (x) u^{'} (x) + y_{2} (x) v^{'} (x)) + y_{1} (x) u^{''} (x) + 2 u^{'} (x) y_{1}^{'} (x) + y_{2} (x) v^{''} (x) + 2 v^{'} (x) y_{2}^{'} (x)$

In[18]:=

rhs[u[x] y1[x] + v[x] y2[x],x,p,q]

$p (x) {(u (x) y1 (x) + v (x) y2 (x))}^{'} (x) + q (x) (u (x) y1 (x) + v (x) y2 (x)) (x) + {(u (x) y1 (x) + v (x) y2 (x))}^{''} (x)$