
Differential Equations Ordinary and Partial Differential Equations Math Forum 
 LinkBack  Thread Tools  Display Modes 
October 4th, 2018, 06:07 AM  #1 
Senior Member Joined: Dec 2015 From: somewhere Posts: 640 Thanks: 91  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, 11:11 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,683 Thanks: 2664 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, 12:03 PM  #3  
Senior Member Joined: Sep 2016 From: USA Posts: 645 Thanks: 408 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
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  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Nontrivial solution of a homogeneous system  jones123  Algebra  11  June 21st, 2016 01:22 AM 
Particular Solution for homogeneous case  JohnofGaunt  Differential Equations  2  June 7th, 2014 06:42 AM 
help need for finding nonhomogeneous solution  skvashok  Applied Math  1  December 21st, 2012 06:05 AM 
find particular solution of nonhomogeneous  mbradar2  Calculus  3  October 13th, 2010 08:49 PM 
Nontrivial solution of a homogeneous system  jones123  Calculus  2  December 31st, 1969 04:00 PM 