60.5.36 problem 1571
Internal
problem
ID
[11570]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
4,
linear
fourth
order
Problem
number
:
1571
Date
solved
:
Tuesday, January 28, 2025 at 06:06:54 PM
CAS
classification
:
[[_high_order, _with_linear_symmetries]]
\begin{align*} \nu ^{4} x^{4} y^{\prime \prime \prime \prime }+\left (4 \nu -2\right ) \nu ^{3} x^{3} y^{\prime \prime \prime }+\left (\nu -1\right ) \left (2 \nu -1\right ) \nu ^{2} x^{2} y^{\prime \prime }-\frac {b^{4} x^{\frac {2}{\nu }} y}{16}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.878 (sec). Leaf size: 143
dsolve(nu^4*x^4*diff(diff(diff(diff(y(x),x),x),x),x)+(4*nu-2)*nu^3*x^3*diff(diff(diff(y(x),x),x),x)+(nu-1)*(2*nu-1)*nu^2*x^2*diff(diff(y(x),x),x)-1/16*b^4*x^(2/nu)*y(x)=0,y(x), singsol=all)
\[
y = \sqrt {x}\, \left (\operatorname {BesselJ}\left (\frac {1}{{\lfloor \frac {1}{\nu }\rfloor }}, \frac {\sqrt {\frac {b^{2}}{\nu ^{2}}}\, x^{\frac {{\lfloor \frac {1}{\nu }\rfloor }}{2}}}{{\lfloor \frac {1}{\nu }\rfloor }}\right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{{\lfloor \frac {1}{\nu }\rfloor }}, \frac {\sqrt {\frac {b^{2}}{\nu ^{2}}}\, x^{\frac {{\lfloor \frac {1}{\nu }\rfloor }}{2}}}{{\lfloor \frac {1}{\nu }\rfloor }}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {1}{{\lfloor \frac {1}{\nu }\rfloor }}, \frac {\sqrt {-\frac {b^{2}}{\nu ^{2}}}\, x^{\frac {{\lfloor \frac {1}{\nu }\rfloor }}{2}}}{{\lfloor \frac {1}{\nu }\rfloor }}\right ) c_3 +\operatorname {BesselY}\left (\frac {1}{{\lfloor \frac {1}{\nu }\rfloor }}, \frac {\sqrt {-\frac {b^{2}}{\nu ^{2}}}\, x^{\frac {{\lfloor \frac {1}{\nu }\rfloor }}{2}}}{{\lfloor \frac {1}{\nu }\rfloor }}\right ) c_4 \right )
\]
✓ Solution by Mathematica
Time used: 0.045 (sec). Leaf size: 195
DSolve[-1/16*(b^4*x^(2/\[Nu])*y[x]) + (-1 + \[Nu])*\[Nu]^2*(-1 + 2*\[Nu])*x^2*D[y[x],{x,2}] + \[Nu]^3*(-2 + 4*\[Nu])*x^3*D[y[x],{x,3}] + \[Nu]^4*x^4*D[y[x],{x,4}] == 0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to 8^{-\nu -1} b^{\nu } \left (x^{2/\nu }\right )^{\nu /4} \left (4^{\nu } (4 c_1 \operatorname {Gamma}(1-\nu )-i c_2 \operatorname {Gamma}(2-\nu )) \operatorname {BesselJ}\left (-\nu ,b \sqrt [4]{x^{2/\nu }}\right )+4^{\nu } (4 c_1 \operatorname {Gamma}(1-\nu )+i c_2 \operatorname {Gamma}(2-\nu )) \operatorname {BesselI}\left (-\nu ,b \sqrt [4]{x^{2/\nu }}\right )+i^{\nu } \left ((4 c_3 \operatorname {Gamma}(\nu +1)-i c_4 \operatorname {Gamma}(\nu +2)) \operatorname {BesselJ}\left (\nu ,b \sqrt [4]{x^{2/\nu }}\right )+(4 c_3 \operatorname {Gamma}(\nu +1)+i c_4 \operatorname {Gamma}(\nu +2)) \operatorname {BesselI}\left (\nu ,b \sqrt [4]{x^{2/\nu }}\right )\right )\right )
\]