76.13.4 problem Ex. 4

Internal problem ID [20089]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter III. Equations of the first order but not of the first degree. Problems at page 32
Problem number : Ex. 4
Date solved : Thursday, October 02, 2025 at 05:22:36 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{3}&=a \,x^{4} \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 62

ode:=diff(y(x),x)^3 = a*x^4; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {3 x^{2} \left (a x \right )^{{1}/{3}}}{7}+c_1 \\ y &= -\frac {3 \left (1+i \sqrt {3}\right ) x^{2} \left (a x \right )^{{1}/{3}}}{14}+c_1 \\ y &= \frac {3 \left (i \sqrt {3}-1\right ) x^{2} \left (a x \right )^{{1}/{3}}}{14}+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 74

ode=D[y[x],x]^3==a*x^4; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {3}{7} \sqrt [3]{a} x^{7/3}+c_1\\ y(x)&\to -\frac {3}{7} \sqrt [3]{-1} \sqrt [3]{a} x^{7/3}+c_1\\ y(x)&\to \frac {3}{7} (-1)^{2/3} \sqrt [3]{a} x^{7/3}+c_1 \end{align*}

✓ Sympy. Time used: 0.643 (sec). Leaf size: 92

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*x**4 + Derivative(y(x), x)**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = C_{1} + \frac {3 \sqrt [3]{a} x^{\frac {7}{3}}}{7}, \ y{\left (x \right )} = C_{1} - \frac {3 \sqrt [3]{a} x^{\frac {7}{3}}}{14} - \frac {3 \sqrt {3} i \sqrt [3]{a} x^{\frac {7}{3}}}{14}, \ y{\left (x \right )} = C_{1} - \frac {3 \sqrt [3]{a} x^{\frac {7}{3}}}{14} + \frac {3 \sqrt {3} i \sqrt [3]{a} x^{\frac {7}{3}}}{14}\right ] \]