76.40.5 problem Ex. 5

Internal problem ID [20236]
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 : Friday, October 03, 2025 at 07:38:10 AM
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: 0.229 (sec). Leaf size: 87

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]
 

\begin{align*} y(x)&\to \frac {1}{45} e^{-\frac {4 x^{3/2}}{3}} x \left (9 e^{\frac {4 x^{3/2}}{3}}+20 (-1)^{2/3} \sqrt [3]{6} c_1 \Gamma \left (-\frac {2}{3},-\frac {4 x^{3/2}}{3}\right )+10 \sqrt [3]{6} \left (3 c_2-2 (-1)^{2/3} c_1 \operatorname {Gamma}\left (-\frac {2}{3}\right )\right )\right ) \end{align*}

✗ 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