60.5.16 problem 1551

Internal problem ID [11550]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1551
Date solved : Tuesday, January 28, 2025 at 06:06:49 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime \prime \prime }-2 \left (\nu ^{2} x^{2}+6\right ) y^{\prime \prime }+\nu ^{2} \left (\nu ^{2} x^{2}+4\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 62

dsolve(x^2*diff(y(x),x$4)-2*(nu^2*x^2+6)*diff(y(x),x$2)+nu^2*(nu^2*x^2+4)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {\left (c_4 \,\nu ^{2} x^{3}+6 c_4 \nu \,x^{2}+15 c_4 x +c_{2} \right ) {\mathrm e}^{-\nu x}+{\mathrm e}^{\nu x} \left (x^{3} \nu ^{2} c_3 -6 x^{2} \nu c_3 +15 x c_3 +c_{1} \right )}{x} \]

✓ Solution by Mathematica

Time used: 0.431 (sec). Leaf size: 181

DSolve[x^2*D[y[x],{x,4}]-2*(nu^2*x^2+6)*D[y[x],{x,2}]+nu^2*(nu^2*x^2+4)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {c_4 x \exp \left (\int \frac {\nu ^3 x \left (x^3-1\right )+\nu ^2 \left (-4 x^3+6 x+1\right )+3 \nu \left (3 x^2-5 x-2\right )+15}{(x-1) x \left (\nu ^2 \left (x^2+x+1\right )-6 \nu (x+1)+15\right )} \, dx\right )+c_3 x \exp \left (\int \frac {\nu ^3 \left (x-x^4\right )+\nu ^2 \left (-4 x^3+6 x+1\right )+\nu \left (-9 x^2+15 x+6\right )+15}{(x-1) x \left (\nu ^2 \left (x^2+x+1\right )+6 \nu (x+1)+15\right )} \, dx\right )+c_1 e^{-\nu x-1}+c_2 e^{\nu x}}{x} \]