15.3.20 problem 20

Internal problem ID [2913]
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 : 20
Date solved : Monday, January 27, 2025 at 06:54:03 AM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} 2 x +y+\left (4 x -2 y+1\right ) y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (\frac {1}{2}\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 21.511 (sec). Leaf size: 84

dsolve([(2*x+y(x))+(4*x-2*y(x)+1)*diff(y(x),x)=0,y(1/2) = 0],y(x), singsol=all)
 

\[ y = \operatorname {RootOf}\left (6 \sqrt {41}\, \operatorname {arctanh}\left (\frac {\left (32 \textit {\_Z} -13-40 x \right ) \sqrt {41}}{328 x +41}\right )+41 \ln \left (\frac {16 \textit {\_Z}^{2}-40 x \textit {\_Z} -16 x^{2}-13 \textit {\_Z} +6 x +2}{\left (8 x +1\right )^{2}}\right )+82 \ln \left (8 x +1\right )+6 \sqrt {41}\, \operatorname {arctanh}\left (\frac {33 \sqrt {41}}{205}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.176 (sec). Leaf size: 128

DSolve[{(2*x+y[x])+(4*x-2*y[x]+1)*D[y[x],x]==0,{y[1/2]==0}},y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\frac {9}{656} \left (6 \sqrt {41} \text {arctanh}\left (\frac {-\frac {2 (8 x+1)}{-2 y(x)+4 x+1}-3}{\sqrt {41}}\right )+41 \left (\log \left (\frac {2 \left (16 x^2-16 y(x)^2+(40 x+13) y(x)-6 x-2\right )}{(8 x+1)^2}\right )+2 \log (8 x+1)\right )\right )=\frac {1}{656} \left (-9 \left (6 \sqrt {41} \text {arctanh}\left (\frac {19}{3 \sqrt {41}}\right )-82 \log (5)+41 \log \left (\frac {25}{2}\right )\right )+369 i \pi \right ),y(x)\right ] \]