36.1.11 problem 11

Internal problem ID [6266]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.2, Separable Equations. Exercises. page 46
Problem number : 11
Date solved : Wednesday, March 05, 2025 at 12:28:02 AM
CAS classification : [_separable]

\begin{align*} x v^{\prime }&=\frac {1-4 v^{2}}{3 v} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 43

ode:=x*diff(v(x),x) = 1/3*(1-4*v(x)^2)/v(x); 
dsolve(ode,v(x), singsol=all);
 

\begin{align*} v \left (x \right ) &= -\frac {\sqrt {x^{{8}/{3}} \left (x^{{8}/{3}}+4 c_{1} \right )}}{2 x^{{8}/{3}}} \\ v \left (x \right ) &= \frac {\sqrt {x^{{8}/{3}} \left (x^{{8}/{3}}+4 c_{1} \right )}}{2 x^{{8}/{3}}} \\ \end{align*}

✓ Mathematica. Time used: 1.919 (sec). Leaf size: 67

ode=x*D[v[x],x]==(1-4*v[x]^2)/(3*v[x]); 
ic={}; 
DSolve[{ode,ic},v[x],x,IncludeSingularSolutions->True]
 

\begin{align*} v(x)\to -\frac {1}{2} \sqrt {1+\frac {e^{8 c_1}}{x^{8/3}}} \\ v(x)\to \frac {1}{2} \sqrt {1+\frac {e^{8 c_1}}{x^{8/3}}} \\ v(x)\to -\frac {1}{2} \\ v(x)\to \frac {1}{2} \\ \end{align*}

✓ Sympy. Time used: 0.544 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
v = Function("v") 
ode = Eq(x*Derivative(v(x), x) - (1 - 4*v(x)**2)/(3*v(x)),0) 
ics = {} 
dsolve(ode,func=v(x),ics=ics)
 

\[ \left [ v{\left (x \right )} = - \frac {\sqrt {\frac {C_{1}}{x^{\frac {8}{3}}} + 1}}{2}, \ v{\left (x \right )} = \frac {\sqrt {\frac {C_{1}}{x^{\frac {8}{3}}} + 1}}{2}\right ] \]