Converting a Second-Order IVP into a System of Equations: Can Substitution Help?

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The discussion focuses on converting the second-order initial value problem (IVP) represented by the equation $y'' + y' - 2y = 0$ with initial conditions $y(0) = 2$ and $y'(0) = 0$ into a system of first-order equations. The substitution method is employed, where $u = y'$ and consequently $y'' = u'$. This leads to the reformulated equation $u' + u - 2y = 0$, simplifying the analysis of the system. The transformation effectively reduces the complexity of solving the IVP.

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  • Understanding of second-order differential equations
  • Familiarity with initial value problems (IVPs)
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karush
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Change the second-order IVP into a system of equations
$y''+y'-2y=0 \quad y(0)= 2\quad y'(0)=0$

let $u=y'$

ok I stuck on this substitution stuff
 
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Since y'= u, y''= u' so y''+ y'- 2y= u'+ u- 2y= 0 so u'= 2y-u.
 

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