(Solution of Cox-Ingersoll-Ross model). This exercise solves the ordinary differential equations $(6.5 .14)$ and (6.5.15) to produce the solutions $C(t, T)$ and $A(t, T)$ given in (6.5.16) and (6.5.17).
(i) Define the function
$$
\varphi(t)=\exp \left\{\frac{1}{2} \sigma^2 \int_t^T C(u, T) d u\right\} .
$$
Show that
$$
\begin{aligned}
C(t, T) & =-\frac{2 \varphi^{\prime}(t)}{\sigma^2 \varphi(t)}, \\
C^{\prime}(t, T) & =-\frac{2 \varphi^{\prime \prime}(t)}{\sigma^2 \varphi(t)}+\frac{1}{2} \sigma^2 C^2(t, T) .
\end{aligned}
$$
(ii) Use the equation (6.5.14) to show that
$$
\varphi^{\prime \prime}(t)-b \varphi^{\prime}(t)-\frac{1}{2} \sigma^2 \varphi(t)=0 .
$$
This is a constant-coefficient linear ordinary differential equation. All solutions are of the form
$$
\varphi(t)=a_1 e^{\lambda_1 t}+a_2 e^{\lambda_2 t},
$$
where $\lambda_1$ and $\lambda_2$ are solutions of the so-called characteristic equation $\lambda^2-$ $b \lambda-\frac{1}{2} \sigma^2=0$, and $a_1$ and $a_2$ are constants.
(iii) Show that $\varphi(t)$ must be of the form
$$
\varphi(t)=\frac{c_1}{\frac{1}{2} b+\gamma} e^{-\left(\frac{1}{2} b+\gamma\right)(T-t)}-\frac{c_2}{\frac{1}{2} b-\gamma} e^{-\left(\frac{1}{2} b-\gamma\right)(T-t)}
$$
for some constants $c_1$ and $c_2$, where $\gamma=\frac{1}{2} \sqrt{b^2+2 \sigma^2}$.