15.1.29 problem 29

Internal problem ID [2869]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 5, page 21
Problem number : 29
Date solved : Monday, January 27, 2025 at 06:21:07 AM
CAS classification : [_separable]

\begin{align*} \left (x^{2}+x +1\right ) y^{\prime }&=y^{2}+2 y+5 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.172 (sec). Leaf size: 35

dsolve([(x^2+x+1)*diff(y(x),x)=y(x)^2+2*y(x)+5,y(1) = 1],y(x), singsol=all)
 

\[ y = -1+2 \cot \left (-\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 x +1\right )}{3}\right )}{3}+\frac {4 \pi \sqrt {3}}{9}+\frac {\pi }{4}\right ) \]

✓ Solution by Mathematica

Time used: 0.822 (sec). Leaf size: 44

DSolve[{(x^2+x+1)*D[y[x],x]==y[x]^2+2*y[x]+5,y[1]==1},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to 2 \tan \left (\frac {4 \arctan \left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{36} \left (9-16 \sqrt {3}\right ) \pi \right )-1 \]