 # Convert the equation into a first-order linear differential equation system with an appropriate transformation of variables.y"+4y'+3y=x^2 Sinead Mcgee 2021-09-06 Answered

Convert the equation into a first-order linear differential equation system with an appropriate transformation of variables.
$y"+4{y}^{\prime }+3y={x}^{2}$

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$y"+4{y}^{\prime }+3y={x}^{2}\dots \left(1\right)$
Let $y={u}_{1}\dots \left(2\right)$
${y}^{\prime }={u}_{2}\dots \left(3\right)$
From (2) , ${y}^{\prime }={u}_{1}^{\prime }⇒{u}_{1}^{\prime }={u}_{2}$
From (3) , $y{u}_{2}^{\prime }$
$⇒{u}_{2}^{\prime }=y$
From (1) , $y{x}^{2}-4{y}^{\prime }-3y$
$={x}^{2}-4{u}_{2}-3{u}_{1}$
$⇒{u}_{2}^{\prime }=-3{u}_{1}-4{u}_{2}+{x}^{2}$
Hence ${u}_{1}^{\prime }={u}_{2}$
${u}_{2}^{\prime }=-3{u}_{1}-4{u}_{2}+{x}^{2}$
$⇒\left[\begin{array}{c}{u}_{1}^{\prime }\\ {u}_{2}^{\prime }\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -3& -4\end{array}\right]\left[\begin{array}{c}{u}_{1}\\ {u}_{2}\end{array}\right]+\left[\begin{array}{c}0\\ {x}^{2}\end{array}\right]$
$⇒{v}^{\prime }=Av+b$
Where $u=\left[\begin{array}{c}{u}_{1}\\ {u}_{2}\end{array}\right],A=\left[\begin{array}{cc}0& 1\\ -3& -4\end{array}\right],b=\left[\begin{array}{c}0\\ {x}^{2}\end{array}\right]$