70.12.7 problem 7

Internal problem ID [18850]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.2 (Theory of second order linear homogeneous equations). Problems at page 226
Problem number : 7
Date solved : Thursday, October 02, 2025 at 03:32:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

With initial conditions

\begin{align*} y \left (0\right )&=y_{0} \\ y^{\prime }\left (0\right )&=y_{1} \\ \end{align*}

✓ Maple. Time used: 0.049 (sec). Leaf size: 56

ode:=(-x^2+1)*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+alpha*(alpha+1)*mu^2/(-x^2+1)*y(x) = 0; 
ic:=[y(0) = y__0, D(y)(0) = y__1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {y_{0} \cos \left (\mu \sqrt {\alpha }\, \sqrt {\alpha +1}\, \operatorname {arctanh}\left (x \right )\right ) \sqrt {\alpha }\, \sqrt {\alpha +1}\, \mu +y_{1} \sin \left (\mu \sqrt {\alpha }\, \sqrt {\alpha +1}\, \operatorname {arctanh}\left (x \right )\right )}{\sqrt {\alpha }\, \sqrt {\alpha +1}\, \mu } \]

✓ Mathematica. Time used: 4.158 (sec). Leaf size: 288

ode=(1-x^2)*D[y[x],{x,2}]-2*x*D[y[x],x]+(a*(a+1)*u^2/(1-x^2))*y[x]==0; 
ic={y[0]==y0,Derivative[1][y][0]==y1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {i \exp \left (\int _1^x\frac {K[1]+\sqrt {-a (a+1) u^2}}{K[1]^2-1}dK[1]-\int _1^0\frac {K[1]+\sqrt {-a (a+1) u^2}}{K[1]^2-1}dK[1]\right ) \left (-\left (\text {y0} \sqrt {-a (a+1) u^2}+\text {y1}\right ) \exp \left (2 \int _1^0\frac {K[1]+\sqrt {-a (a+1) u^2}}{K[1]^2-1}dK[1]\right ) \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {K[1]+\sqrt {-a (a+1) u^2}}{K[1]^2-1}dK[1]\right )dK[2]+\left (\text {y0} \sqrt {-a (a+1) u^2}+\text {y1}\right ) \exp \left (2 \int _1^0\frac {K[1]+\sqrt {-a (a+1) u^2}}{K[1]^2-1}dK[1]\right ) \int _1^0\exp \left (-2 \int _1^{K[2]}\frac {K[1]+\sqrt {-a (a+1) u^2}}{K[1]^2-1}dK[1]\right )dK[2]-\text {y0}\right )}{\sqrt {x^2-1}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
Alpha = symbols("Alpha") 
mu = symbols("mu") 
y = Function("y") 
ode = Eq(Alpha*mu**2*(Alpha + 1)*y(x)/(1 - x**2) - 2*x*Derivative(y(x), x) + (1 - x**2)*Derivative(y(x), (x, 2)),0) 
ics = {y(0): y__0, Subs(Derivative(y(x), x), x, 0): y__1} 
dsolve(ode,func=y(x),ics=ics)
 

False