Internal problem ID [3611]
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]
Solve \begin {gather*} \boxed {\left (5+3 x \right ) \left (y^{\prime }\right )^{2}-\left (3+3 y\right ) y^{\prime }+y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.203 (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 x y \relax (x )+9 y \relax (x )^{2}-2 y \relax (x )+9}-18 x +27 y \relax (x )-3\right ) {\mathrm e}^{-\frac {3 \left (3 y \relax (x )+3+\sqrt {-12 x y \relax (x )+9 y \relax (x )^{2}-2 y \relax (x )+9}\right )}{2 \left (5+3 x \right )}} c_{1}}{5+3 x}+x -\frac {\left (27 \expIntegral \left (1, -\frac {3 \left (3 y \relax (x )+3+\sqrt {-12 x y \relax (x )+9 y \relax (x )^{2}-2 y \relax (x )+9}\right )}{2 \left (5+3 x \right )}\right ) \sqrt {-12 x y \relax (x )+9 y \relax (x )^{2}-2 y \relax (x )+9}-54 \expIntegral \left (1, -\frac {3 \left (3 y \relax (x )+3+\sqrt {-12 x y \relax (x )+9 y \relax (x )^{2}-2 y \relax (x )+9}\right )}{2 \left (5+3 x \right )}\right ) x +81 \expIntegral \left (1, -\frac {3 \left (3 y \relax (x )+3+\sqrt {-12 x y \relax (x )+9 y \relax (x )^{2}-2 y \relax (x )+9}\right )}{2 \left (5+3 x \right )}\right ) y \relax (x )+24 \,{\mathrm e}^{\frac {\frac {9 y \relax (x )}{2}+\frac {9}{2}+\frac {3 \sqrt {-12 x y \relax (x )+9 y \relax (x )^{2}-2 y \relax (x )+9}}{2}}{5+3 x}} x -9 \expIntegral \left (1, -\frac {3 \left (3 y \relax (x )+3+\sqrt {-12 x y \relax (x )+9 y \relax (x )^{2}-2 y \relax (x )+9}\right )}{2 \left (5+3 x \right )}\right )+40 \,{\mathrm e}^{\frac {\frac {9 y \relax (x )}{2}+\frac {9}{2}+\frac {3 \sqrt {-12 x y \relax (x )+9 y \relax (x )^{2}-2 y \relax (x )+9}}{2}}{5+3 x}}\right ) {\mathrm e}^{-\frac {3 \left (3 y \relax (x )+3+\sqrt {-12 x y \relax (x )+9 y \relax (x )^{2}-2 y \relax (x )+9}\right )}{2 \left (5+3 x \right )}}}{6 \left (5+3 x \right )} = 0 \\ \frac {\left (9 \sqrt {-12 x y \relax (x )+9 y \relax (x )^{2}-2 y \relax (x )+9}+18 x -27 y \relax (x )+3\right ) {\mathrm e}^{\frac {-\frac {9 y \relax (x )}{2}-\frac {9}{2}+\frac {3 \sqrt {-12 x y \relax (x )+9 y \relax (x )^{2}-2 y \relax (x )+9}}{2}}{5+3 x}} c_{1}}{5+3 x}+x +\frac {\left (27 \expIntegral \left (1, \frac {-\frac {9 y \relax (x )}{2}-\frac {9}{2}+\frac {3 \sqrt {-12 x y \relax (x )+9 y \relax (x )^{2}-2 y \relax (x )+9}}{2}}{5+3 x}\right ) \sqrt {-12 x y \relax (x )+9 y \relax (x )^{2}-2 y \relax (x )+9}+54 \expIntegral \left (1, \frac {-\frac {9 y \relax (x )}{2}-\frac {9}{2}+\frac {3 \sqrt {-12 x y \relax (x )+9 y \relax (x )^{2}-2 y \relax (x )+9}}{2}}{5+3 x}\right ) x -81 \expIntegral \left (1, \frac {-\frac {9 y \relax (x )}{2}-\frac {9}{2}+\frac {3 \sqrt {-12 x y \relax (x )+9 y \relax (x )^{2}-2 y \relax (x )+9}}{2}}{5+3 x}\right ) y \relax (x )-24 \,{\mathrm e}^{-\frac {3 \left (-3 y \relax (x )-3+\sqrt {-12 x y \relax (x )+9 y \relax (x )^{2}-2 y \relax (x )+9}\right )}{2 \left (5+3 x \right )}} x +9 \expIntegral \left (1, \frac {-\frac {9 y \relax (x )}{2}-\frac {9}{2}+\frac {3 \sqrt {-12 x y \relax (x )+9 y \relax (x )^{2}-2 y \relax (x )+9}}{2}}{5+3 x}\right )-40 \,{\mathrm e}^{-\frac {3 \left (-3 y \relax (x )-3+\sqrt {-12 x y \relax (x )+9 y \relax (x )^{2}-2 y \relax (x )+9}\right )}{2 \left (5+3 x \right )}}\right ) {\mathrm e}^{\frac {-\frac {9 y \relax (x )}{2}-\frac {9}{2}+\frac {3 \sqrt {-12 x y \relax (x )+9 y \relax (x )^{2}-2 y \relax (x )+9}}{2}}{5+3 x}}}{30+18 x} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.488 (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]
\[ \text {Solve}\left [\left \{x=\frac {e^{-3 K[1]} (3 K[1]-1) \left ((9-27 K[1]) \text {ExpIntegralEi}(3 K[1])+4 e^{3 K[1]}\right )}{9 K[1]-3}+c_1 e^{-3 K[1]} (3 K[1]-1),y(x)=\frac {3 x K[1]^2}{3 K[1]-1}+\frac {5 K[1]^2-3 K[1]}{3 K[1]-1}\right \},\{y(x),K[1]\}\right ] \]