60.3.377 problem 1383

Internal problem ID [11387]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1383
Date solved : Monday, January 27, 2025 at 11:18:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 39

dsolve(diff(diff(y(x),x),x) = -((alpha+beta+1)*(x-a)^2*(x-b)+(1-alpha-beta)*(x-b)^2*(x-a))/(x-a)^2/(x-b)^2*diff(y(x),x)-alpha*beta*(a-b)^2/(x-a)^2/(x-b)^2*y(x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 2.903 (sec). Leaf size: 187

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

\[ y(x)\to \exp \left (\int _1^x\frac {-\sqrt {(\alpha -\beta )^2} a+a+b-2 K[1]+b \sqrt {(\alpha -\beta )^2}}{2 (b-K[1]) (K[1]-a)}dK[1]-\frac {1}{2} \int _1^x\left (\frac {\alpha +\beta -1}{a-K[2]}+\frac {\alpha +\beta +1}{K[2]-b}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {-\sqrt {(\alpha -\beta )^2} a+a+b-2 K[1]+b \sqrt {(\alpha -\beta )^2}}{2 (a-K[1]) (K[1]-b)}dK[1]\right )dK[3]+c_1\right ) \]