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Building the Sine Function Term by Term

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In this post we will solve the following basic second-order equation with corresponding initial conditions:

$\displaystyle y'' = -y, \qquad y(0) = 0, \qquad y'(0)=1$

This equation can be solved very easily exactly. The answer is $y(x) = sin(x)$ . Now, let’s pretend that we don’t know how to solve this equation exactly. We introduce a parameter $\epsilon$ and consider the family of equations

$\displaystyle y'' = -\epsilon y$

$\displaystyle y'' +\epsilon y = 0$

and the function $y(x)$ in the form:

$\displaystyle y_{\epsilon}(x) = a_0(x) + a_1(x)\epsilon + a_2(x)\epsilon^2 + a_3(x)\epsilon^3 + ...$

$\displaystyle y_{\epsilon}''(x) = a_0''(x) + a_1''(x)\epsilon + a_2''(x)\epsilon^2 + a_3''(x)\epsilon^3 + ...$

The differential equation becomes (up to three terms):

$\displaystyle y'' +\epsilon y = 0$

$\displaystyle a_0''(x) + a_1''(x)\epsilon + a_2''(x)\epsilon^2 + a_3''(x)\epsilon^3 + \epsilon (a_0(x) + a_1(x)\epsilon + a_2(x)\epsilon^2 + a_3(x)\epsilon^3) = 0$