60.3.292 problem 1297

Internal problem ID [11302]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1297
Date solved : Monday, January 27, 2025 at 11:05:45 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 57

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

\[ y = c_{1} \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+1}\right )^{\frac {i \sqrt {b}}{\sqrt {a}}}+c_{2} \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+1}\right )^{-\frac {i \sqrt {b}}{\sqrt {a}}} \]

✓ Solution by Mathematica

Time used: 0.044 (sec). Leaf size: 52

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

\[ y(x)\to c_1 \cos \left (\frac {\sqrt {b} \text {arcsinh}\left (\sqrt {a} x\right )}{\sqrt {a}}\right )+c_2 \sin \left (\frac {\sqrt {b} \text {arcsinh}\left (\sqrt {a} x\right )}{\sqrt {a}}\right ) \]