81.2.6 problem 6
Internal
problem
ID
[18614]
Book
:
A
short
course
on
differential
equations.
By
Donald
Francis
Campbell.
Maxmillan
company.
London.
1907
Section
:
Chapter
II.
Change
of
variable.
Exercises
at
page
20
Problem
number
:
6
Date
solved
:
Wednesday, January 29, 2025 at 03:39:52 AM
CAS
classification
:
[[_3rd_order, _with_linear_symmetries]]
\begin{align*} y^{2} y^{\prime \prime \prime }-\left (3 y^{\prime } y+2 x y^{2}\right ) y^{\prime \prime }+\left (2 {y^{\prime }}^{2}+2 x y y^{\prime }+3 x^{2} y^{2}\right ) y^{\prime }+x^{3} y^{3}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.267 (sec). Leaf size: 556
dsolve(y(x)^2*diff(y(x),x$3)-(3*y(x)*diff(y(x),x)+2*x*y(x)^2 )*diff(y(x),x$2)+(2*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)+3*x^2*y(x)^2)*diff(y(x),x)+x^3*y(x)^3=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= 0 \\
y \left (x \right ) &= {\mathrm e}^{-\frac {233 \left (\left (\int x^{4} {\mathrm e}^{-\frac {x^{2} \left (i \sqrt {2}+1\right )}{2}} \operatorname {KummerU}\left (\frac {3}{4}+\frac {i \sqrt {2}}{8}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )d x \right ) \left (\int x \operatorname {KummerM}\left (\frac {3}{4}+\frac {i \sqrt {2}}{8}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) {\mathrm e}^{-\frac {x^{2} \left (i \sqrt {2}-1\right )}{2}}d x \right )-\left (\int x^{4} {\mathrm e}^{-\frac {x^{2} \left (i \sqrt {2}+1\right )}{2}} \operatorname {KummerM}\left (\frac {3}{4}+\frac {i \sqrt {2}}{8}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )d x \right ) \left (\int x \operatorname {KummerU}\left (\frac {3}{4}+\frac {i \sqrt {2}}{8}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) {\mathrm e}^{-\frac {x^{2} \left (i \sqrt {2}-1\right )}{2}}d x \right )-c_{1} \left (\int x \operatorname {KummerU}\left (\frac {3}{4}+\frac {i \sqrt {2}}{8}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) {\mathrm e}^{-\frac {x^{2} \left (i \sqrt {2}-1\right )}{2}}d x \right )+\left (\int x \operatorname {KummerM}\left (\frac {3}{4}+\frac {i \sqrt {2}}{8}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) {\mathrm e}^{-\frac {x^{2} \left (i \sqrt {2}-1\right )}{2}}d x \right ) c_{2} +\int x^{4} {\mathrm e}^{-\frac {x^{2} \left (i \sqrt {2}+1\right )}{2}} \operatorname {KummerM}\left (\frac {3}{4}+\frac {i \sqrt {2}}{8}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \left (\int x \operatorname {KummerU}\left (\frac {3}{4}+\frac {i \sqrt {2}}{8}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) {\mathrm e}^{-\frac {x^{2} \left (i \sqrt {2}-1\right )}{2}}d x \right )d x -\int x^{4} {\mathrm e}^{-\frac {x^{2} \left (i \sqrt {2}+1\right )}{2}} \operatorname {KummerU}\left (\frac {3}{4}+\frac {i \sqrt {2}}{8}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \left (\int x \operatorname {KummerM}\left (\frac {3}{4}+\frac {i \sqrt {2}}{8}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) {\mathrm e}^{-\frac {x^{2} \left (i \sqrt {2}-1\right )}{2}}d x \right )d x +c_3 \right ) \left (\frac {5104 \operatorname {KummerU}\left (\frac {i \sqrt {2}}{8}-\frac {1}{4}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \left (i \sqrt {2}-\frac {599}{638}\right ) \operatorname {KummerM}\left (\frac {3}{4}+\frac {i \sqrt {2}}{8}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )}{4427}+\operatorname {KummerU}\left (\frac {3}{4}+\frac {i \sqrt {2}}{8}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \left (i \sqrt {2}-\frac {122}{233}\right ) \operatorname {KummerM}\left (\frac {i \sqrt {2}}{8}-\frac {1}{4}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )\right ) {\mathrm e}^{i \sqrt {2}\, x^{2}}}{152 x {\left (\operatorname {KummerU}\left (\frac {i \sqrt {2}}{8}-\frac {1}{4}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \left (\frac {29 i \sqrt {2}}{19}+\frac {22}{19}\right ) \operatorname {KummerM}\left (\frac {3}{4}+\frac {i \sqrt {2}}{8}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )+\left (i \sqrt {2}+\frac {5}{4}\right ) \operatorname {KummerU}\left (\frac {3}{4}+\frac {i \sqrt {2}}{8}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right ) \operatorname {KummerM}\left (\frac {i \sqrt {2}}{8}-\frac {1}{4}, \frac {3}{2}, i \sqrt {2}\, x^{2}\right )\right )}^{2}}} \\
\end{align*}
✗ Solution by Mathematica
Time used: 0.000 (sec). Leaf size: 0
DSolve[y[x]^2*D[y[x],{x,3}]-(3*y[x]*D[y[x],x]+2*x*y[x]^2 )*D[y[x],{x,2}]+(2*D[y[x],x]^2+2*x*y[x]*D[y[x],x]+3*x^2*y[x]^2)*D[y[x],x]+x^3*y[x]^3==0,y[x],x,IncludeSingularSolutions -> True]
Not solved