81.3.17 problem 17

Internal problem ID [18635]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter III. Ordinary differential equations of the first order and first degree. Exercises at page 33
Problem number : 17
Date solved : Tuesday, January 28, 2025 at 12:05:06 PM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 82

dsolve(y(x)*(3+y(x))*diff(y(x),x)=x*(2*y(x)+3),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\frac {3 \,{\mathrm e}^{-\frac {4 x^{2}}{9}-\frac {1}{2}-\frac {8 c_{1}}{9}}}{2}-\frac {3 \sqrt {-\frac {{\mathrm e}^{-1-\frac {8 x^{2}}{9}-\frac {16 c_{1}}{9}}}{\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-1-\frac {8 x^{2}}{9}-\frac {16 c_{1}}{9}}}{9}\right )}}}{2}}{\sqrt {-\frac {{\mathrm e}^{-1-\frac {8 x^{2}}{9}-\frac {16 c_{1}}{9}}}{\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-1-\frac {8 x^{2}}{9}-\frac {16 c_{1}}{9}}}{9}\right )}}} \]

✓ Solution by Mathematica

Time used: 0.362 (sec). Leaf size: 48

DSolve[y[x]*(3+y[x])*D[y[x],x]==x*(2*y[x]+3),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {1}{16} (2 \text {$\#$1}+3)^2-\frac {9}{8} \log (2 \text {$\#$1}+3)\&\right ]\left [\frac {x^2}{2}+c_1\right ] \\ y(x)\to -\frac {3}{2} \\ \end{align*}