69.14.12 problem 338

Internal problem ID [18213]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 14. Differential equations admitting of depression of their order. Exercises page 107
Problem number : 338
Date solved : Thursday, October 02, 2025 at 03:09:11 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_poly_yn]]

\begin{align*} 2 y^{\prime \prime }&=\frac {y^{\prime }}{x}+\frac {x^{2}}{y^{\prime }} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=\frac {\sqrt {2}}{5} \\ y^{\prime }\left (1\right )&=\frac {\sqrt {2}}{2} \\ \end{align*}

✓ Maple. Time used: 0.096 (sec). Leaf size: 12

ode:=2*diff(diff(y(x),x),x) = diff(y(x),x)/x+x^2/diff(y(x),x); 
ic:=[y(1) = 1/5*2^(1/2), D(y)(1) = 1/2*2^(1/2)]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\sqrt {2}\, x^{{5}/{2}}}{5} \]

✓ Mathematica. Time used: 0.059 (sec). Leaf size: 26

ode=2*D[y[x],{x,2}]==D[y[x],x]/x+x^2/D[y[x],x]; 
ic={y[1]==Sqrt[2]/5,Derivative[1][y][1]==Sqrt[2]/2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{5} \sqrt {2} x^{3/2} \sqrt {x^2} \end{align*}

✓ Sympy. Time used: 40.034 (sec). Leaf size: 78

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2/Derivative(y(x), x) + 2*Derivative(y(x), (x, 2)) - Derivative(y(x), x)/x,0) 
ics = {y(1): sqrt(2)/5, Subs(Derivative(y(x), x), x, 1): sqrt(2)/2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {\sqrt {2} \int \sqrt {x^{3}}\, dx}{2} - \frac {\sqrt {2} \int \limits ^{1} \sqrt {x^{3}}\, dx}{2} + \frac {\sqrt {2}}{5}, \ y{\left (x \right )} = \frac {\sqrt {2} \int \sqrt {x^{3}}\, dx}{2} - \frac {\sqrt {2} \int \limits ^{1} \sqrt {x^{3}}\, dx}{2} + \frac {\sqrt {2}}{5}\right ] \]