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If $p$ and $r$ in $y^{\prime}+p(x) y=r(x)$ are continuous for all $x$ in an interval $\left|x-x_{0}\right|<a,$ show that $f(x, y)$ in this 0 DE satisfies the con...

Question

If $p$ and $r$ in $y^{\prime}+p(x) y=r(x)$ are continuous for all $x$ in an interval $\left|x-x_{0}\right|<a,$ show that $f(x, y)$ in this 0 DE satisfies the conditions of our present theorems, so that a corresponding initial value problem has a unique solution. Do you actually need these theorems for this ODE?

If $p$ and $r$ in $y^{\prime}+p(x) y=r(x)$ are continuous for all $x$ in an interval $\left|x-x_{0}\right|<a,$ show that $f(x, y)$ in this 0 DE satisfies the conditions of our present theorems, so that a corresponding initial value problem has a unique solution. Do you actually need these theorems for this ODE?



Answers

Second-order linear differential equations take the form

$$y^{\prime \prime}(t)+p(t) y^{\prime}(t)+q(t) y(t)=g(t)$$

where $p, q,$ and $g$ are continuous functions of $t .$ Suppose we have initial conditions $y(0)=a$ and $y^{\prime}(0)=b .$ Show that this equation can be rewritten as a system of two first-order linear differential equations having the form

$\frac{d \mathbf{x}}{d t}=\left[ \begin{array}{cc}{0} & {1} \\ {-q(t)} & {-p(t)}\end{array}\right] \mathbf{x}+\left[ \begin{array}{c}{0} \\ {g(t)}\end{array}\right]$

$\begin{aligned} \text { with } & \mathbf{x}(0)=\left[ \begin{array}{l}{a} \\ {b}\end{array}\right] \\ \text { where } x_{1}(t)=y(t) \text { and } x_{2}(t) &=y^{\prime}(t) \end{aligned}$


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