67.1.29 problem 2.4 (g)

Internal problem ID [16294]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 2. Integration and differential equations. Additional exercises. page 32
Problem number : 2.4 (g)
Date solved : Thursday, October 02, 2025 at 10:45:26 AM
CAS classification : [[_2nd_order, _quadrature]]

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

With initial conditions

\begin{align*} y \left (1\right )&=8 \\ y^{\prime }\left (1\right )&=6 \\ \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 19

ode:=x*diff(diff(y(x),x),x)+2 = x^(1/2); 
ic:=[y(1) = 8, D(y)(1) = 6]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {4 x^{{3}/{2}}}{3}-2 x \ln \left (x \right )+6 x +\frac {2}{3} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 26

ode=x*D[y[x],{x,2}]+2==Sqrt[x]; 
ic={y[1]==8,Derivative[1][y][1]==6}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {2}{3} \left (2 x^{3/2}+9 x-3 x \log (x)+1\right ) \end{align*}

✓ Sympy. Time used: 0.120 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sqrt(x) + x*Derivative(y(x), (x, 2)) + 2,0) 
ics = {y(1): 8, Subs(Derivative(y(x), x), x, 1): 6} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {4 x^{\frac {3}{2}}}{3} - 2 x \log {\left (x \right )} + 6 x + \frac {2}{3} \]