60.4.75 problem 1525
Internal
problem
ID
[11526]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
3,
linear
third
order
Problem
number
:
1525
Date
solved
:
Tuesday, January 28, 2025 at 06:06:45 PM
CAS
classification
:
[[_3rd_order, _with_linear_symmetries]]
\begin{align*} x^{6} y^{\prime \prime \prime }+6 x^{5} y^{\prime \prime }+a y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.033 (sec). Leaf size: 204
dsolve(x^6*diff(diff(diff(y(x),x),x),x)+6*x^5*diff(diff(y(x),x),x)+a*y(x)=0,y(x), singsol=all)
\[
y = \frac {4 \left (-8 x^{3}+a \right )^{4} \left (c_3 \,{\mathrm e}^{\frac {\left (-a^{4}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2 a x}} \left (\frac {\left (-i+\sqrt {3}\right ) \left (-a^{4}\right )^{{1}/{3}}}{4}+i a x \right )+\left (\frac {\left (\sqrt {3}+i\right ) \left (-a^{4}\right )^{{1}/{3}}}{4}-i a x \right ) c_{2} {\mathrm e}^{-\frac {\left (-a^{4}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2 a x}}+128 c_{1} {\mathrm e}^{-\frac {\left (-a^{4}\right )^{{1}/{3}}}{a x}} \left (a x +\frac {\left (-a^{4}\right )^{{1}/{3}}}{2}\right )\right )}{{\left (2 a x +\left (-a^{4}\right )^{{1}/{3}}\right )}^{4} {\left (i \left (-a^{4}\right )^{{1}/{3}} \sqrt {3}-4 a x +\left (-a^{4}\right )^{{1}/{3}}\right )}^{4} {\left (i \left (-a^{4}\right )^{{1}/{3}} \sqrt {3}+4 a x -\left (-a^{4}\right )^{{1}/{3}}\right )}^{4}}
\]
✓ Solution by Mathematica
Time used: 0.517 (sec). Leaf size: 165
DSolve[a*y[x] + 6*x^5*D[y[x],{x,2}] + x^6*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to c_1 \exp \left (-\int \frac {a^{2/3}-2 \sqrt [3]{a} x+2 x^2}{x^2 \left (\sqrt [3]{a}-2 x\right )} \, dx\right )+c_2 \exp \left (\int \frac {\sqrt [3]{-1} a^{2/3}+2 (-1)^{2/3} \sqrt [3]{a} x-2 x^2}{x^2 \left ((-1)^{2/3} \sqrt [3]{a}-2 x\right )} \, dx\right )+c_3 \exp \left (\int \frac {(-1)^{2/3} a^{2/3}+2 \sqrt [3]{-1} \sqrt [3]{a} x+2 x^2}{x^2 \left (\sqrt [3]{-1} \sqrt [3]{a}+2 x\right )} \, dx\right )
\]