74.6.14 problem 10 (c)

Internal problem ID [19864]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter IV. Methods of solution: First order equations. section 33. Problems at page 91
Problem number : 10 (c)
Date solved : Thursday, October 02, 2025 at 04:51:37 PM
CAS classification : [_separable]

\begin{align*} u \ln \left (u \right ) v^{\prime }+\sin \left (v\right )^{2}&=1 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 10

ode:=u*ln(u)*diff(v(u),u)+sin(v(u))^2 = 1; 
dsolve(ode,v(u), singsol=all);
 

\[ v = \arctan \left (\ln \left (\ln \left (u \right )\right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.288 (sec). Leaf size: 52

ode=u*Log[u]*D[v[u],u]+Sin[v[u]]==0; 
ic={}; 
DSolve[{ode,ic},v[u],u,IncludeSingularSolutions->True]
 

\begin{align*} v(u)&\to -\arccos (-\tanh (-\log (\log (u))+c_1))\\ v(u)&\to \arccos (-\tanh (-\log (\log (u))+c_1))\\ v(u)&\to 0\\ v(u)&\to -\pi \\ v(u)&\to \pi \end{align*}

✓ Sympy. Time used: 1.017 (sec). Leaf size: 76

from sympy import * 
u = symbols("u") 
v = Function("v") 
ode = Eq(u*log(u)*Derivative(v(u), u) + sin(v(u))**2 - 1,0) 
ics = {} 
dsolve(ode,func=v(u),ics=ics)
 

\[ \left [ v{\left (u \right )} = 2 \operatorname {atan}{\left (\frac {\sqrt {C_{1}^{2} + 2 C_{1} \log {\left (\log {\left (u \right )} \right )} + \log {\left (\log {\left (u \right )} \right )}^{2} + 1} - 1}{C_{1} + \log {\left (\log {\left (u \right )} \right )}} \right )}, \ v{\left (u \right )} = - 2 \operatorname {atan}{\left (\frac {\sqrt {C_{1}^{2} + 2 C_{1} \log {\left (\log {\left (u \right )} \right )} + \log {\left (\log {\left (u \right )} \right )}^{2} + 1} + 1}{C_{1} + \log {\left (\log {\left (u \right )} \right )}} \right )}\right ] \]