60.5.37 problem 1572

Internal problem ID [11571]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1572
Date solved : Tuesday, January 28, 2025 at 06:06:55 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-1\right )^{2} y^{\prime \prime \prime \prime }+10 x \left (x^{2}-1\right ) y^{\prime \prime \prime }+\left (24 x^{2}-8-2 \left (\mu \left (\mu +1\right )+\nu \left (\nu +1\right )\right ) \left (x^{2}-1\right )\right ) y^{\prime \prime }-6 x \left (\mu \left (\mu +1\right )+\nu \left (\nu +1\right )-2\right ) y^{\prime }+\left (\left (\mu \left (\mu +1\right )-\nu \left (\nu +1\right )\right )^{2}-2 \mu \left (\mu +1\right )-2 \nu \left (\nu +1\right )\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.300 (sec). Leaf size: 35

dsolve((x^2-1)^2*diff(diff(diff(diff(y(x),x),x),x),x)+10*x*(x^2-1)*diff(diff(diff(y(x),x),x),x)+(24*x^2-8-2*(mu*(mu+1)+nu*(nu+1))*(x^2-1))*diff(diff(y(x),x),x)-6*x*(mu*(mu+1)+nu*(nu+1)-2)*diff(y(x),x)+((mu*(mu+1)-nu*(nu+1))^2-2*mu*(mu+1)-2*nu*(nu+1))*y(x)=0,y(x), singsol=all)
 

\[ y = \left (\operatorname {LegendreQ}\left (\mu , x\right ) c_{2} +c_{1} \operatorname {LegendreP}\left (\mu , x\right )\right ) \operatorname {LegendreP}\left (\nu , x\right )+\operatorname {LegendreQ}\left (\nu , x\right ) \left (\operatorname {LegendreQ}\left (\mu , x\right ) c_4 +\operatorname {LegendreP}\left (\mu , x\right ) c_3 \right ) \]

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

DSolve[(-2*\[Mu]*(1 + \[Mu]) - 2*\[Nu]*(1 + \[Nu]) + (\[Mu]*(1 + \[Mu]) - \[Nu]*(1 + \[Nu]))^2)*y[x] - 6*(-2 + \[Mu]*(1 +\[Mu]) + \[Nu]*(1 + \[Nu]))*x*D[y[x],x] + (-8 + 24*x^3 - 2*(\[Mu]*(1 + \[Mu]) + \[Nu]*(1 + \[Nu]))*(-1 + x^2))*D[y[x],{x,2}] + 10*x*(-1 + x^2)*Derivative[3][y][x] + (-1 + x^2)^2*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

Not solved