Reduction of order

Reduction of order is a process used to find the solution of a differential equation $y''+p(t)y'+q(t)y=0$, when Euler substitution methods find only one value for $\lambda$. (That is, ${\sqrt {b^{2}-4ac}}=0$ and $\lambda ={\frac {-b}{2a}}$.)

The process is carried out in the following manner:

1. $y_{1}=e^{\lambda t}$. However, we must find $y_{2}$. $y_{2}$ cannot be a multiple of $y_{1}$, so we assume that $y_{2}=v(t)y_{1}$.

2. We apply the product rule to find:

$y_{2}=v(t)y_{1}$

$y_{2}'=v(t)'y_{1}+v(t)y_{1}'$

$y_{2}''=v(t)''y_{1}+2v(t)'y_{1}'+v(t)y_{1}''$

3. We substitute these expressions into the initial differential equation:

$y_{2}''+p(t)y_{2}'+q(t)y=(v(t)''y_{1}+2v(t)'y_{1}'+v(t)y_{1}'')+p(t)(v(t)'y_{1}+v(t)y_{1}')+q(t)(v(t)y_{1})$

4. We collect the $v(t)''$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(t)'} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(t)} terms:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(t)''(y_1) + v(t)'(2y_1' + p(t) y_1) + v(t)(y_1'' + p(t) y_1' + q(t) v(t) y_1)}

5. The terms Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2y_1' + p(t) y_1)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (y_1'' + p(t) y_1' + q(t) v(t) y_1)} equal zero in most instances, leading to the conclusion:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(t)''(y_1) = 0}

Integrating twice, we yield:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(t) = c_1 t + c_2}

6. This is enough to say that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_2 = v(t) y_1 = ty_1 = t e^{\lambda t}}

7. The solution is then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = c_1 y_1 + c_2 y_2 = e^{\lambda t} (c_1 + c_2 t)}