Internal problem ID [4119]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 30
Problem number: 883.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_rational, _dAlembert]
\[ \boxed {\left (5+3 x \right ) {y^{\prime }}^{2}-\left (3+3 y\right ) y^{\prime }+y=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 745
dsolve((5+3*x)*diff(y(x),x)^2-(3+3*y(x))*diff(y(x),x)+y(x) = 0,y(x), singsol=all)
\begin{align*} \frac {\left (9 \sqrt {-12 y \left (x \right ) x +9 y \left (x \right )^{2}-2 y \left (x \right )+9}-18 x +27 y \left (x \right )-3\right ) {\mathrm e}^{-\frac {3 \left (3+3 y \left (x \right )+\sqrt {-12 y \left (x \right ) x +9 y \left (x \right )^{2}-2 y \left (x \right )+9}\right )}{2 \left (5+3 x \right )}} c_{1}}{5+3 x}+x -\frac {\left (27 \,\operatorname {Ei}_{1}\left (-\frac {3 \left (3+3 y \left (x \right )+\sqrt {-12 y \left (x \right ) x +9 y \left (x \right )^{2}-2 y \left (x \right )+9}\right )}{2 \left (5+3 x \right )}\right ) \sqrt {-12 y \left (x \right ) x +9 y \left (x \right )^{2}-2 y \left (x \right )+9}-54 \,\operatorname {Ei}_{1}\left (-\frac {3 \left (3+3 y \left (x \right )+\sqrt {-12 y \left (x \right ) x +9 y \left (x \right )^{2}-2 y \left (x \right )+9}\right )}{2 \left (5+3 x \right )}\right ) x +81 \,\operatorname {Ei}_{1}\left (-\frac {3 \left (3+3 y \left (x \right )+\sqrt {-12 y \left (x \right ) x +9 y \left (x \right )^{2}-2 y \left (x \right )+9}\right )}{2 \left (5+3 x \right )}\right ) y \left (x \right )+24 \,{\mathrm e}^{\frac {\frac {9}{2}+\frac {9 y \left (x \right )}{2}+\frac {3 \sqrt {-12 y \left (x \right ) x +9 y \left (x \right )^{2}-2 y \left (x \right )+9}}{2}}{5+3 x}} x -9 \,\operatorname {Ei}_{1}\left (-\frac {3 \left (3+3 y \left (x \right )+\sqrt {-12 y \left (x \right ) x +9 y \left (x \right )^{2}-2 y \left (x \right )+9}\right )}{2 \left (5+3 x \right )}\right )+40 \,{\mathrm e}^{\frac {\frac {9}{2}+\frac {9 y \left (x \right )}{2}+\frac {3 \sqrt {-12 y \left (x \right ) x +9 y \left (x \right )^{2}-2 y \left (x \right )+9}}{2}}{5+3 x}}\right ) {\mathrm e}^{-\frac {3 \left (3+3 y \left (x \right )+\sqrt {-12 y \left (x \right ) x +9 y \left (x \right )^{2}-2 y \left (x \right )+9}\right )}{2 \left (5+3 x \right )}}}{6 \left (5+3 x \right )} = 0 \frac {\left (9 \sqrt {-12 y \left (x \right ) x +9 y \left (x \right )^{2}-2 y \left (x \right )+9}+18 x -27 y \left (x \right )+3\right ) {\mathrm e}^{\frac {-\frac {9}{2}-\frac {9 y \left (x \right )}{2}+\frac {3 \sqrt {-12 y \left (x \right ) x +9 y \left (x \right )^{2}-2 y \left (x \right )+9}}{2}}{5+3 x}} c_{1}}{5+3 x}+x +\frac {\left (27 \,\operatorname {Ei}_{1}\left (\frac {-\frac {9}{2}-\frac {9 y \left (x \right )}{2}+\frac {3 \sqrt {-12 y \left (x \right ) x +9 y \left (x \right )^{2}-2 y \left (x \right )+9}}{2}}{5+3 x}\right ) \sqrt {-12 y \left (x \right ) x +9 y \left (x \right )^{2}-2 y \left (x \right )+9}+54 \,\operatorname {Ei}_{1}\left (\frac {-\frac {9}{2}-\frac {9 y \left (x \right )}{2}+\frac {3 \sqrt {-12 y \left (x \right ) x +9 y \left (x \right )^{2}-2 y \left (x \right )+9}}{2}}{5+3 x}\right ) x -81 \,\operatorname {Ei}_{1}\left (\frac {-\frac {9}{2}-\frac {9 y \left (x \right )}{2}+\frac {3 \sqrt {-12 y \left (x \right ) x +9 y \left (x \right )^{2}-2 y \left (x \right )+9}}{2}}{5+3 x}\right ) y \left (x \right )-24 \,{\mathrm e}^{-\frac {3 \left (-3-3 y \left (x \right )+\sqrt {-12 y \left (x \right ) x +9 y \left (x \right )^{2}-2 y \left (x \right )+9}\right )}{2 \left (5+3 x \right )}} x +9 \,\operatorname {Ei}_{1}\left (\frac {-\frac {9}{2}-\frac {9 y \left (x \right )}{2}+\frac {3 \sqrt {-12 y \left (x \right ) x +9 y \left (x \right )^{2}-2 y \left (x \right )+9}}{2}}{5+3 x}\right )-40 \,{\mathrm e}^{-\frac {3 \left (-3-3 y \left (x \right )+\sqrt {-12 y \left (x \right ) x +9 y \left (x \right )^{2}-2 y \left (x \right )+9}\right )}{2 \left (5+3 x \right )}}\right ) {\mathrm e}^{\frac {-\frac {9}{2}-\frac {9 y \left (x \right )}{2}+\frac {3 \sqrt {-12 y \left (x \right ) x +9 y \left (x \right )^{2}-2 y \left (x \right )+9}}{2}}{5+3 x}}}{30+18 x} = 0 \end{align*}
✓ Solution by Mathematica
Time used: 1.709 (sec). Leaf size: 106
DSolve[(5+3 x) (y'[x])^2-(3+3 y[x])y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=\frac {e^{-3 K[1]} (3 K[1]-1) \left ((9-27 K[1]) \operatorname {ExpIntegralEi}(3 K[1])+4 e^{3 K[1]}\right )}{9 K[1]-3}+c_1 e^{-3 K[1]} (3 K[1]-1),y(x)=\frac {3 x K[1]^2}{3 K[1]-1}+\frac {5 K[1]^2-3 K[1]}{3 K[1]-1}\right \},\{y(x),K[1]\}\right ] \]