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 October 4th, 2018, 07:07 AM #1 Senior Member   Joined: Dec 2015 From: Earth Posts: 817 Thanks: 113 Math Focus: Elementary Math Homogeneous solution explain Given equation $\displaystyle y'+py=q$ for $\displaystyle q=0$ then $\displaystyle y=y_h$ is the homogeneous solution (1) Explain why solution to equation is $\displaystyle y=y_h+y_p$ where $\displaystyle y_h$ - homogeneous and $\displaystyle y_p$ -particular (2) Explain how to derive $\displaystyle y_p$ from $\displaystyle y_h$ How to derive $\displaystyle y=y_h + y_p$ ? October 4th, 2018, 12:11 PM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,696 Thanks: 2681 Math Focus: Mainly analysis and algebra 1) Suppose that $y_p$ is any solution of $y'+py=q$ and that $y_c$ is any solution of $y'+py=0$. Then if $y= y_c+y_p$, we have $y'=y_c'+y_p'$ and so \begin{align}y'+py &= (y_c'+y_p') + p(y_c+y_p) \\ &= (y_c' + py_c) + (y_p' + py_p) &(\text{just grouping the terms differently}) \\ &= 0 + q \\ &= q\end{align} October 4th, 2018, 01:03 PM   #3
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 Originally Posted by v8archie 1) Suppose that $y_p$ is any solution of $y'+py=q$ and that $y_c$ is any solution of $y'+py=0$. Then if $y= y_c+y_p$, we have $y'=y_c'+y_p'$ and so \begin{align}y'+py &= (y_c'+y_p') + p(y_c+y_p) \\ &= (y_c' + py_c) + (y_p' + py_p) &(\text{just grouping the terms differently}) \\ &= 0 + q \\ &= q\end{align}
This is true but only gets you one direction. He/she still has to show that if $x$ is any solution to this ODE, then it has the required form. For this it suffices to show that the homogenous equation has a unique solution (via another application of v8Archie's argument).

For this the usual approach is to assume write the equation as $x' = Ax$ so that $\exp(tA)$ is a solution. Now assume that $x$ is a solution and and show that $\exp(tA)x(t)$ is constant.

For the second part, this is just the variation of constants formula:
$x(t) = \exp(tA) \left( \exp(-t_0A)x_0 + \int_{t_0}^{t} \exp(-sA) g(s) \ ds \right)$ Tags explain, homogeneous, solution Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post jones123 Algebra 11 June 21st, 2016 02:22 AM JohnofGaunt Differential Equations 2 June 7th, 2014 07:42 AM skvashok Applied Math 1 December 21st, 2012 07:05 AM mbradar2 Calculus 3 October 13th, 2010 09:49 PM jones123 Calculus 2 December 31st, 1969 04:00 PM

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