60.4.64 problem 1514

Internal problem ID [11515]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1514
Date solved : Monday, January 27, 2025 at 11:22:50 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime \prime }+6 x^{2} y^{\prime \prime }+\left (x^{3} a -12\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.029 (sec). Leaf size: 129

dsolve(x^3*diff(diff(diff(y(x),x),x),x)+6*x^2*diff(diff(y(x),x),x)+(a*x^3-12)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {-c_{2} \left (\left (-i-\sqrt {3}\right ) \left (-a^{4}\right )^{{2}/{3}}+i a^{3} x \right ) {\mathrm e}^{-\frac {\left (-a^{4}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right ) x}{2 a}}-\left (\left (-i+\sqrt {3}\right ) \left (-a^{4}\right )^{{2}/{3}}+i a^{3} x \right ) c_3 \,{\mathrm e}^{\frac {\left (-a^{4}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right ) x}{2 a}}+c_{1} \left (a^{3} x +2 \left (-a^{4}\right )^{{2}/{3}}\right ) {\mathrm e}^{\frac {\left (-a^{4}\right )^{{1}/{3}} x}{a}}}{x^{3}} \]

✓ Solution by Mathematica

Time used: 0.541 (sec). Leaf size: 167

DSolve[(-12 + a*x^3)*y[x] + 6*x^2*D[y[x],{x,2}] + x^3*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \exp \left (-\int \frac {a^{2/3} x^2+4 \sqrt [3]{a} x+6}{\sqrt [3]{a} x^2+2 x} \, dx\right )+c_2 \exp \left (\int \frac {\sqrt [3]{-1} a^{2/3} x^2-4 \sqrt [3]{a} x-6 (-1)^{2/3}}{x \left (\sqrt [3]{a} x+2 (-1)^{2/3}\right )} \, dx\right )+c_3 \exp \left (\int \frac {-(-1)^{2/3} a^{2/3} x^2-4 \sqrt [3]{a} x+6 \sqrt [3]{-1}}{x \left (\sqrt [3]{a} x-2 \sqrt [3]{-1}\right )} \, dx\right ) \]