23.3.71 problem 73

Internal problem ID [5785]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 73
Date solved : Tuesday, September 30, 2025 at 02:03:02 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} \csc \left (a \right )^{2} y-2 \tan \left (a \right ) y^{\prime }+y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 69

ode:=csc(a)^2*y(x)-2*tan(a)*diff(y(x),x)+diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_2 \,{\mathrm e}^{2 \sec \left (a \right )^{2} \sqrt {\cos \left (a \right )^{6}-3 \cos \left (a \right )^{4}+\cos \left (a \right )^{2}}\, \csc \left (a \right ) x}+c_1 \right ) {\mathrm e}^{-x \left (-\tan \left (a \right )+\sqrt {\cos \left (a \right )^{6}-3 \cos \left (a \right )^{4}+\cos \left (a \right )^{2}}\, \sec \left (a \right )^{2} \csc \left (a \right )\right )} \]

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 58

ode=Csc[a]^2*y[x] - 2*Tan[a]*D[y[x],x] + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 e^{x \left (\tan (a)-\sqrt {\tan ^2(a)-\csc ^2(a)}\right )}+c_2 e^{x \left (\tan (a)+\sqrt {\tan ^2(a)-\csc ^2(a)}\right )} \end{align*}

✓ Sympy. Time used: 0.266 (sec). Leaf size: 70

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(y(x)/sin(a)**2 - 2*tan(a)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} e^{x \left (- \frac {\sqrt {\left (\frac {\sin ^{4}{\left (a \right )}}{\cos ^{2}{\left (a \right )}} - 1\right ) \sin ^{4}{\left (a \right )}}}{\sin ^{3}{\left (a \right )}} + \tan {\left (a \right )}\right )} + C_{2} e^{x \left (\frac {\sqrt {\left (\frac {\sin ^{4}{\left (a \right )}}{\cos ^{2}{\left (a \right )}} - 1\right ) \sin ^{4}{\left (a \right )}}}{\sin ^{3}{\left (a \right )}} + \tan {\left (a \right )}\right )} \]