====== Math 527 ungraded variation of parameters homework ======

===== problem 5 solution =====

(a) What property must an operator $L$ satisfy to be linear? 

\begin{eqnarray*} 
L (c_1 y_1 + c_2 y_2) = c_1 L y_1 + c_2 L_y_2
\end{eqnarray*}

for all constants $c_1, c_2$ and all functions $y_1(x), y_2(x)$.

(b) Why is linearity important for the solution of linear differential equations?

Because it allows you to express the general solution of the ODE as a sum of 
the other solutions. 

%%(c)%% How many linearly independent solutions does an $n$th order linear homogeneous equation have?

$n$

(d) When you integrate $u_1'$ and $u_2'$ in variation of parameters, why can you always set the 
integration constant to zero? 

Because $u_1$ and $u_2$ are coefficients of the homogeneous solutions $y_1$ and $y_2$  in the ansatz

\begin{eqnarray*}
y_p = u_1 y_1 + u_2 y_2
\end{eqnarray*}

any constant
included in the value of $u_1$ or $u_2$ or could just be absorbed into the constants in front of 
$y_1$ and $y_2$ in the general solution. E.g.

\begin{eqnarray*}
c_1 y_1 + c_2 y_2 + y_p &= c_1 y_1 + c_2 y_2 + (u_1 + a) y_1 + (u_2 + b) y_2 \\
                        &= (c_1 + a) y_1 + (c_2 + a) y_2 + u_1 y_1 + u_2 y_2 \\
\end{eqnarray*}


(e) What is Euler's formula?

\begin{eqnarray*}
e^{ix} = \cos x + i \sin x
\end{eqnarray*}

(f) How would you prove Euler's formula? Don't do the proof, just describe the proof in a sentence or two.

Substitute $ix$ in place of $x$ in the power series  expansion of $e^x$, then simplify and regroup so that the
even terms become the power series for $\cos x$ and the odd terms become $i$ times the power series for $\sin x$.