60.4.48 problem 1496

Internal problem ID [11499]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1496
Date solved : Tuesday, January 28, 2025 at 06:06:38 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime \prime }+6 x y^{\prime \prime }+6 y^{\prime }+a \,x^{2} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 55

dsolve(x^2*diff(diff(diff(y(x),x),x),x)+6*x*diff(diff(y(x),x),x)+6*diff(y(x),x)+a*x^2*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {c_{1} {\mathrm e}^{\frac {\left (-a \right )^{{1}/{3}} \left (i \sqrt {3}-1\right ) x}{2}}+c_{2} {\mathrm e}^{-\frac {\left (-a \right )^{{1}/{3}} \left (1+i \sqrt {3}\right ) x}{2}}+c_3 \,{\mathrm e}^{\left (-a \right )^{{1}/{3}} x}}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.160 (sec). Leaf size: 62

DSolve[a*x^2*y[x] + 6*D[y[x],x] + 6*x*D[y[x],{x,2}] + x^2*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {c_1 e^{-\sqrt [3]{a} x-2}+c_2 e^{\sqrt [3]{-1} \sqrt [3]{a} x}+c_3 e^{-(-1)^{2/3} \sqrt [3]{a} x-2}}{x^2} \]