77.1.110 problem 138 (page 198)

Internal problem ID [18000]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 138 (page 198)
Date solved : Tuesday, January 28, 2025 at 11:19:40 AM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.112 (sec). Leaf size: 62

dsolve((1-x^2)*diff(y(x),x$3)-x*diff(y(x),x$2)+diff(y(x),x)=0,y(x), singsol=all)
 

\[ y = \frac {c_{2} x^{2} \sqrt {x^{2}-1}-c_{3} x^{3}+\sqrt {x^{2}-1}\, \ln \left (x +\sqrt {x^{2}-1}\right ) c_{3} +c_{1} \sqrt {x^{2}-1}+c_{3} x}{\sqrt {x^{2}-1}} \]

✓ Solution by Mathematica

Time used: 60.085 (sec). Leaf size: 79

DSolve[(1-x^2)*D[y[x],{x,3}]-x*D[y[x],{x,2}]+D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \int _1^x\left (c_1 \cosh \left (\frac {\arcsin (K[1]) \sqrt {1-K[1]^2}}{\sqrt {K[1]^2-1}}\right )+i c_2 \sinh \left (\frac {\arcsin (K[1]) \sqrt {1-K[1]^2}}{\sqrt {K[1]^2-1}}\right )\right )dK[1]+c_3 \]