15.3.16 problem 16

Internal problem ID [2909]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 7, page 28
Problem number : 16
Date solved : Tuesday, March 04, 2025 at 03:27:12 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} 3 x -y+2+\left (x +2 y+1\right ) y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \end{align*}

✗ Maple

ode:=3*x-y(x)+2+(x+2*y(x)+1)*diff(y(x),x) = 0; 
ic:=y(0) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.189 (sec). Leaf size: 111

ode=(3*x-y[x]+2)+(x+2*y[x]+1)*D[y[x],x]==0; 
ic={y[0]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [2 \sqrt {6} \arctan \left (\frac {\sqrt {\frac {2}{3}} (-y(x)+3 x+2)}{2 y(x)+x+1}\right )=3 \left (\frac {2}{3} \left (\sqrt {6} \arctan \left (2 \sqrt {\frac {2}{3}}\right )+3 \log \left (\frac {25}{22}\right )-6 \log (5)\right )+2 \log \left (\frac {42 x^2+28 y(x)^2+8 y(x)+60 x+22}{(7 x+5)^2}\right )+4 \log (7 x+5)\right ),y(x)\right ] \]

✓ Sympy. Time used: 5.884 (sec). Leaf size: 87

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

\[ \log {\left (x + \frac {5}{7} \right )} = - \log {\left (\sqrt {\frac {3}{2} + \frac {\left (y{\left (x \right )} + \frac {1}{7}\right )^{2}}{\left (x + \frac {5}{7}\right )^{2}}} \right )} - \frac {\sqrt {6} \operatorname {atan}{\left (\frac {\sqrt {6} \left (y{\left (x \right )} + \frac {1}{7}\right )}{3 \left (x + \frac {5}{7}\right )} \right )}}{6} - \log {\left (10 \right )} - \log {\left (7 \right )} + \frac {\sqrt {6} \operatorname {atan}{\left (\frac {\sqrt {6}}{15} \right )}}{6} + \log {\left (5 \right )} + \frac {\log {\left (154 \right )}}{2} \]