67.8.7 problem 13.2 (a)

Internal problem ID [16502]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 13. Higher order equations: Extending first order concepts. Additional exercises page 259
Problem number : 13.2 (a)
Date solved : Thursday, October 02, 2025 at 01:35:38 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} y^{\prime \prime }&=4 x \sqrt {y^{\prime }} \end{align*}

✓ Maple. Time used: 0.562 (sec). Leaf size: 51

ode:=diff(diff(y(x),x),x) = 4*x*diff(y(x),x)^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= c_{1} \\ y &= \frac {x^{5}}{5}-\frac {2 x^{3}}{3 c_{1}}+\frac {x}{c_{1}^{2}}+c_{2} \\ y &= \frac {x^{5}}{5}+\frac {2 x^{3}}{3 c_{1}}+\frac {x}{c_{1}^{2}}+c_{2} \\ \end{align*}

✓ Mathematica. Time used: 0.046 (sec). Leaf size: 33

ode=D[y[x],{x,2}]==4*x*Sqrt[D[y[x],x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x^5}{5}+\frac {c_1 x^3}{3}+\frac {c_1{}^2 x}{4}+c_2 \end{align*}

✓ Sympy. Time used: 0.331 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x*sqrt(Derivative(y(x), x)) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + \frac {C_{2}^{2} x}{4} - \frac {C_{2} x^{3}}{3} + \frac {x^{5}}{5} \]