60.5.15 problem 1550
Internal
problem
ID
[11549]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
4,
linear
fourth
order
Problem
number
:
1550
Date
solved
:
Tuesday, January 28, 2025 at 06:06:48 PM
CAS
classification
:
[[_high_order, _with_linear_symmetries]]
\begin{align*} x y^{\prime \prime \prime \prime }-\left (6 x^{2}+1\right ) y^{\prime \prime \prime }+12 x^{3} y^{\prime \prime }-\left (9 x^{2}-7\right ) x^{2} y^{\prime }+2 \left (x^{2}-3\right ) x^{3} y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.527 (sec). Leaf size: 157
dsolve(x*diff(diff(diff(diff(y(x),x),x),x),x)-(6*x^2+1)*diff(diff(diff(y(x),x),x),x)+12*x^3*diff(diff(y(x),x),x)-(9*x^2-7)*x^2*diff(y(x),x)+2*(x^2-3)*x^3*y(x)=0,y(x), singsol=all)
\[
y = -{\mathrm e}^{x^{2}} \left (\int \frac {\operatorname {WhittakerM}\left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{-\frac {x^{2}}{4}}}{x^{{3}/{2}}}d x \right ) c_3 -{\mathrm e}^{x^{2}} \left (\int \frac {\operatorname {WhittakerW}\left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{-\frac {x^{2}}{4}}}{x^{{3}/{2}}}d x \right ) c_4 +\left (\int \frac {\operatorname {WhittakerM}\left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{x^{{3}/{2}}}d x \right ) {\mathrm e}^{\frac {x^{2}}{2}} c_3 +{\mathrm e}^{\frac {x^{2}}{2}} \left (\int \frac {\operatorname {WhittakerW}\left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{x^{{3}/{2}}}d x \right ) c_4 +{\mathrm e}^{x^{2}} c_{1} +c_{2} {\mathrm e}^{\frac {x^{2}}{2}}
\]
✓ Solution by Mathematica
Time used: 1.211 (sec). Leaf size: 220
DSolve[2*x^3*(-3 + x^2)*y[x] - x^2*(-7 + 9*x^2)*D[y[x],x] + 12*x^3*D[y[x],{x,2}] - (1 + 6*x^2)*Derivative[3][y][x] + x*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to e^{\frac {x^2}{2}} \left (c_3 \int _1^x\frac {e^{\frac {K[1]^2}{2}} \left (\int \frac {e^{\frac {1}{4} \left (-1+\sqrt {5}\right ) K[1]^2-1} \operatorname {HypergeometricU}\left (-\frac {1}{4}+\frac {9}{4 \sqrt {5}},-\frac {1}{2},-\frac {1}{2} \sqrt {5} K[1]^2\right ) \left (K[1]^2\right )^{3/4}}{K[1]^{7/2}} \, dK[1]\right ) K[1]}{\sqrt [4]{2}}dK[1]+c_4 \int _1^x\frac {e^{\frac {K[2]^2}{2}} \left (\int \frac {e^{\frac {1}{4} \left (-1+\sqrt {5}\right ) K[2]^2-1} \left (K[2]^2\right )^{3/4} L_{\frac {1}{4}-\frac {9}{4 \sqrt {5}}}^{-\frac {3}{2}}\left (-\frac {1}{2} \sqrt {5} K[2]^2\right )}{K[2]^{7/2}} \, dK[2]\right ) K[2]}{\sqrt [4]{2}}dK[2]+c_2 e^{\frac {x^2}{2}}+c_1\right )
\]