82.40.5 problem Ex. 5
Internal
problem
ID
[18880]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
VIII.
Exact
differential
equations,
and
equations
of
particular
forms.
Integration
in
series.
problems
at
page
94
Problem
number
:
Ex.
5
Date
solved
:
Monday, March 31, 2025 at 06:18:01 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \sqrt {x}\, y^{\prime \prime }+2 x y^{\prime }+3 y&=x \end{align*}
✓ Maple. Time used: 0.258 (sec). Leaf size: 223
ode:=x^(1/2)*diff(diff(y(x),x),x)+2*x*diff(y(x),x)+3*y(x) = x;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\frac {\left (x^{{3}/{2}}\right )^{{1}/{6}} \Gamma \left (\frac {2}{3}\right ) c_1 \left (4 \Gamma \left (\frac {1}{3}, -\frac {4 x^{{3}/{2}}}{3}\right ) x^{{3}/{2}} {\mathrm e}^{-\frac {4 x^{{3}/{2}}}{3}}+6^{{2}/{3}} \left (-x^{{3}/{2}}\right )^{{1}/{3}}\right )}{x^{{1}/{4}}}-\frac {4 \,6^{{1}/{3}} x^{4} {\mathrm e}^{-\frac {4 x^{{3}/{2}}}{3}} \Gamma \left (\frac {1}{3}, -\frac {4 x^{{3}/{2}}}{3}\right )}{15}+{\mathrm e}^{-\frac {4 x^{{3}/{2}}}{3}} x c_2 \left (-x^{{3}/{2}}\right )^{{1}/{3}}-\frac {8 x^{{5}/{4}} \sqrt {3}\, \pi \,{\mathrm e}^{-\frac {4 x^{{3}/{2}}}{3}} \left (x^{{3}/{2}}\right )^{{1}/{6}} c_1}{3}-\frac {2 x^{{5}/{2}} \left (-x^{{3}/{2}}\right )^{{1}/{3}}}{5}+\frac {{\mathrm e}^{-\frac {4 x^{{3}/{2}}}{3}} x \left (8 \,3^{{5}/{6}} 2^{{1}/{3}} \pi \,x^{3}-5 \int \frac {4 \,3^{{5}/{6}} 2^{{1}/{3}} x^{2} \pi -6 x^{2} 2^{{1}/{3}} 3^{{1}/{3}} \Gamma \left (\frac {1}{3}, -\frac {4 x^{{3}/{2}}}{3}\right ) \Gamma \left (\frac {2}{3}\right )-9 \sqrt {x}\, {\mathrm e}^{\frac {4 x^{{3}/{2}}}{3}} \left (-x^{{3}/{2}}\right )^{{1}/{3}} \Gamma \left (\frac {2}{3}\right )}{\left (-x^{{3}/{2}}\right )^{{1}/{3}}}d x \left (-x^{{3}/{2}}\right )^{{1}/{3}}\right )}{45 \Gamma \left (\frac {2}{3}\right )}}{\left (-x^{{3}/{2}}\right )^{{1}/{3}}}
\]
✓ Mathematica. Time used: 0.493 (sec). Leaf size: 288
ode=Sqrt[x]*D[y[x],{x,2}]+2*x*D[y[x],x]+3*y[x]==x;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {(-2)^{2/3} e^{-\frac {4 x^{3/2}}{3}} x \left (9\ 6^{2/3} e^{\frac {4 x^{3/2}}{3}} \sqrt [3]{-x^{3/2}} \text {Root}\left [9 \text {$\#$1}^3-16\&,3\right ]+48 x^3 \operatorname {Gamma}\left (-\frac {2}{3}\right ) \text {Root}\left [9 \text {$\#$1}^3-16\&,3\right ]-48 x^3 \Gamma \left (-\frac {2}{3},-\frac {4 x^{3/2}}{3}\right ) \text {Root}\left [9 \text {$\#$1}^3-16\&,3\right ]-16 x^3 \operatorname {Gamma}\left (-\frac {2}{3}\right ) \text {Root}\left [\text {$\#$1}^3-48\&,3\right ]+16 x^3 \Gamma \left (-\frac {2}{3},-\frac {4 x^{3/2}}{3}\right ) \text {Root}\left [\text {$\#$1}^3-48\&,3\right ]\right )}{45 \sqrt [3]{3} \sqrt [3]{-x^{3/2}} \text {Root}\left [9 \text {$\#$1}^3-16\&,3\right ] \text {Root}\left [\text {$\#$1}^3-48\&,3\right ]}+\frac {2 \sqrt [3]{2} c_2 e^{-\frac {4 x^{3/2}}{3}} x}{3^{2/3}}-\frac {4 (-1)^{2/3} \sqrt [3]{2} c_1 e^{-\frac {4 x^{3/2}}{3}} x \left (\operatorname {Gamma}\left (-\frac {2}{3}\right )-\Gamma \left (-\frac {2}{3},-\frac {4 x^{3/2}}{3}\right )\right )}{3\ 3^{2/3}}
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(sqrt(x)*Derivative(y(x), (x, 2)) + 2*x*Derivative(y(x), x) - x + 3*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - 1/2 + 3*y(x)/(2*x) + Derivative(y(x), (x, 2))/(2*sqrt(x)) cannot be solved by the factorable group method