83.37.9 problem 9

Internal problem ID [19454]
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 (B) at page 128
Problem number : 9
Date solved : Tuesday, January 28, 2025 at 01:44:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 20

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

\[ y \left (x \right ) = \sec \left (x \right ) \left (c_{1} \sinh \left (a x \right )+c_{2} \cosh \left (a x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.047 (sec). Leaf size: 32

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

\[ y(x)\to \sec (x) \left (c_1 e^{-a x}+\frac {c_2 e^{a x}}{2 a}\right ) \]