38.1.53 problem 64

Internal problem ID [8214]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 1. Introduction to differential equations. Exercises 1.1 at page 12
Problem number : 64
Date solved : Tuesday, September 30, 2025 at 05:18:54 PM
CAS classification : [_quadrature]

\begin{align*} x {y^{\prime }}^{2}-4 y^{\prime }-12 x^{3}&=0 \end{align*}

✓ Maple. Time used: 0.036 (sec). Leaf size: 49

ode:=x*diff(y(x),x)^2-4*diff(y(x),x)-12*x^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -2 \int \frac {-1+\sqrt {3 x^{4}+1}}{x}d x +c_1 \\ y &= 2 \int \frac {1+\sqrt {3 x^{4}+1}}{x}d x +c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 67

ode=x*D[y[x],x]^2-4*D[y[x],x]-12*x^3==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \sqrt {3 x^4+1}+\log \left (\sqrt {3 x^4+1}-1\right )+c_1\\ y(x)&\to -\sqrt {3 x^4+1}+\log \left (\sqrt {3 x^4+1}+1\right )+c_1 \end{align*}

✓ Sympy. Time used: 1.517 (sec). Leaf size: 78

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-12*x**3 + x*Derivative(y(x), x)**2 - 4*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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