$\phi(0)_{M = 2.55} = 0.023$, $\phi(10)_{M = 2.55} = 0$
$\phi(0)_{M = 2.57} = 0.037$, $\phi(12)_{M = 2.57} = 0$
$\phi(0)_{M = 2.55} = 0.046$, $\phi(14)_{M = 2.59} = 0$

$\phi(0)_{M = 2.55} = 0.023$
$\phi(0)_{M = 2.57} = 0.037$
$\phi(0)_{M = 2.55} = 0.046$

In the code below, I first define a function for the first order differential equation. Then, set boundary conditions for each mach number, $M$, and finally, I use odeint from the scipy.integrate module in Python to solve the boundary value problem. The plot of the solutions is shown in the last figure.

Ouput:

In the code below, I first define a function for the first-order differential equation. Then, set boundary conditions for each mach number, $M$, and finally, I use odeint from the scipy.integrate module in Python to solve the boundary value problem. The plot of the solutions is shown in the last figure.

Output:

$\phi(0)_{M = 2.55} = \phi_{0, M = 2.55} = 0.023$
$\phi(0)_{M = 2.57} = \phi_{0, M = 2.57} = 0.037$
$\phi(0)_{M = 2.59} = \phi_{0, M = 2.59} = 0.046$

$\phi(0)_{M = 2.55} = 0.023$, $\phi(10)_{M = 2.55} = 0$
$\phi(0)_{M = 2.57} = 0.037$, $\phi(12)_{M = 2.57} = 0$
$\phi(0)_{M = 2.55} = 0.046$, $\phi(14)_{M = 2.59} = 0$