64.7.14 problem 14

Internal problem ID [13388]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, Section 2.4. Special integrating factors and transformations. Exercises page 67
Problem number : 14
Date solved : Tuesday, January 28, 2025 at 05:36:48 AM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} 4 x +3 y+1+\left (x +y+1\right ) y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (3\right )&=-4 \end{align*}

✓ Solution by Maple

Time used: 0.450 (sec). Leaf size: 39

dsolve([(4*x+3*y(x)+1)+(x+y(x)+1)*diff(y(x),x)=0,y(3) = -4],y(x), singsol=all)
 

\[ y = \frac {-2 x \operatorname {LambertW}\left (-\left (x -2\right ) {\mathrm e}^{-1}\right )+\operatorname {LambertW}\left (-\left (x -2\right ) {\mathrm e}^{-1}\right )-x +2}{\operatorname {LambertW}\left (-\left (x -2\right ) {\mathrm e}^{-1}\right )} \]

✓ Solution by Mathematica

Time used: 0.216 (sec). Leaf size: 165

DSolve[{(4*x+3*y[x]+1)+(x+y[x]+1)*D[y[x],x]==0,{y[-2]==2}},y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _{-2}^{-\frac {(-1)^{2/3} \left (\frac {3 (x-2)}{x+y(x)+1}+2\right )}{\sqrt [3]{2}}}\frac {2}{2 K[1]^3+3 \sqrt [3]{-2} K[1]+2}dK[1]=\frac {1}{9} (-2)^{2/3} \log (x-2)+\frac {1}{9} \left (\frac {1}{2} \left (18 \int _{-2}^{5 (-2)^{2/3}}\frac {2}{2 K[1]^3+3 \sqrt [3]{-2} K[1]+2}dK[1]+2^{2/3} \sqrt {3} \pi +2^{2/3} \log (4)\right )+\frac {1}{2} i \left (2^{2/3} \pi -2^{2/3} \sqrt {3} \log (4)\right )\right ),y(x)\right ] \]