91Ó°ÊÓ

Recall that we can convert Cartesian coordinates \((x, y)\) to polar coordinates \((r, \theta)\) using the following transformation: $$ \begin{aligned} x &= r \cos{(\theta)}, \\ y &= r \sin{(\theta)}. \end{aligned} $$ Additionally, we have: $$ \begin{aligned} x^2 + y^2 &= r^2, \\ \ln(x^2 + y^2) &= \ln(r^2). \end{aligned} $$ Let's differentiate \(x\) and \(y\) with respect to \(t\): $$ \begin{aligned} \frac{dx}{dt} &= r' \cos{(\theta)} - r \sin{(\theta)} \theta', \\ \frac{dy}{dt} &= r' \sin{(\theta)} + r \cos{(\theta)} \theta'. \end{aligned} $$ Substitute the given cartesian IVP into polar coordinate and solve for \(r'\) and \(\theta'\).

We will substitute the polar coordinates transformation results into the Cartesian IVP equations: $$ \begin{aligned} r' \cos{(\theta)} - r \sin{(\theta)} \theta' &= r \sin{(\theta)} - r \cos{(\theta)} \ln(r^2), \\ r' \sin{(\theta)} + r \cos{(\theta)} \theta' &= -r \cos{(\theta)} - r \sin{(\theta)} \ln(r^2). \end{aligned} $$ Now, let's solve these two equations to find expressions for \(r'\) and \(\theta'\): $$ \begin{aligned} r' &= \frac{r (\sin{(\theta)} \cos{(\theta)} + \sin{(\theta)} \cos{(\theta)}\ln{(r^2)})}{\cos^2{(\theta)}+\sin^2{(\theta)}} \\ \theta' &= \frac{-\sin^2{(\theta)} + \cos^2{(\theta)}}{\sin{(\theta)}\cos{(\theta)}} - \ln(r^2). \end{aligned} $$ We can simplify: $$ \begin{aligned} r' &= r (1 - \cos{(\theta)}\sin{(\theta)} \ln{(r^2)}), \\ \theta' &= \cos{(2\theta)} - \ln{(r^2)}. \end{aligned} $$

Convert the initial Cartesian values to polar coordinates: $$ \begin{aligned} r(0) &= \sqrt{(e/\sqrt{2})^2 + (e/\sqrt{2})^2} = e, \\ \theta(0) &= \arctan(\frac{e/\sqrt{2}}{e/\sqrt{2}}) = \frac{\pi}{4}. \end{aligned} $$

Unfortunately, the polar IVP does not have an easy analytical solution. The equations are: $$ \begin{aligned} r' &= r (1 - \cos{(\theta)}\sin{(\theta)} \ln{(r^2)}), \\ \theta' &= \cos{(2\theta)} - \ln{(r^2)}, \end{aligned} $$ with initial conditions $$ \begin{aligned} r(0) &= e, \\ \theta(0) &= \frac{\pi}{4}. \end{aligned} $$ To find the solution we could either use numerical methods or attempt multivariable series expansions. However, neither of these techniques guarantee an analytical solution or the existence of the solution at \(t=1\). Thus, we cannot evaluate the solution at \(t=1\) using the steps in this solution. We would need to employ other techniques, such as numerical methods or advanced series expansions, to further investigate the existence and evaluation of the solution.