83.38.8 problem 8

Internal problem ID [19467]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VIII. Linear equations of second order. Excercise VIII (C) at page 133
Problem number : 8
Date solved : Tuesday, January 28, 2025 at 01:45:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (\tan \left (x \right )-1\right )^{2} y^{\prime }-n \left (n -1\right ) y \sec \left (x \right )^{4}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.387 (sec). Leaf size: 35

dsolve(diff(y(x),x$2)+(tan(x)-1)^2*diff(y(x),x)-n*(n-1)*y(x)*sec(x)^4=0,y(x), singsol=all)
 

\[ y \left (x \right ) = {\mathrm e}^{-\frac {\tan \left (x \right )}{2}} \left (c_{1} \sinh \left (\frac {\tan \left (x \right ) \left (2 n -1\right )}{2}\right )+c_{2} \cosh \left (\frac {\tan \left (x \right ) \left (2 n -1\right )}{2}\right )\right ) \]

✓ Solution by Mathematica

Time used: 1.876 (sec). Leaf size: 209

DSolve[D[y[x],{x,2}]+(Tan[x]-1)^2*D[y[x],x]-n*(n-1)*y[x]*Sec[x]^4==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\frac {\sqrt {-\cos (x)+\sqrt {\cos (x)-1} \sqrt {\cos (x)+1}-1} \sqrt {-\cos (x)+\sqrt {\cos (x)-1} \sqrt {\cos (x)+1}+1} \exp \left (-\frac {1}{2} \sqrt {\cos (x)+1} \sec (x) \left (\sqrt {-(1-2 n)^2} \sqrt {\cos (x)-1}+\sqrt {1-\cos (x)}\right )\right ) \left (c_1 \sqrt {-(1-2 n)^2} \exp \left (\sqrt {-(2 n-1)^2} \sqrt {\cos (x)-1} \sqrt {\cos (x)+1} \sec (x)\right )+c_2\right )}{2 \sqrt {-(1-2 n)^2} \sqrt [4]{-\sin ^2(x)} \left (\sqrt {\cos (x)-1}-\sqrt {\cos (x)+1}\right )} \]