Internal problem ID [3601]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 30
Problem number: 872.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _dAlembert]
Solve \begin {gather*} \boxed {x \left (y^{\prime }\right )^{2}+a y y^{\prime }+b x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.123 (sec). Leaf size: 224
dsolve(x*diff(y(x),x)^2+a*y(x)*diff(y(x),x)+b*x = 0,y(x), singsol=all)
\begin{align*} \frac {\left (-\frac {a \left (-a^{2} y \relax (x )^{2}+\sqrt {a^{2} y \relax (x )^{2}-4 b \,x^{2}}\, a y \relax (x )-a y \relax (x )^{2}+2 b \,x^{2}+\sqrt {a^{2} y \relax (x )^{2}-4 b \,x^{2}}\, y \relax (x )\right )}{2 x^{2}}\right )^{-\frac {a +2}{2 \left (a +1\right )}} \left (-a y \relax (x )+\sqrt {a^{2} y \relax (x )^{2}-4 b \,x^{2}}\right ) c_{1}}{x}+x = 0 \\ \frac {\left (a y \relax (x )+\sqrt {a^{2} y \relax (x )^{2}-4 b \,x^{2}}\right ) c_{1} \left (\frac {a \left (a^{2} y \relax (x )^{2}+\sqrt {a^{2} y \relax (x )^{2}-4 b \,x^{2}}\, a y \relax (x )+a y \relax (x )^{2}-2 b \,x^{2}+\sqrt {a^{2} y \relax (x )^{2}-4 b \,x^{2}}\, y \relax (x )\right )}{2 x^{2}}\right )^{-\frac {a +2}{2 \left (a +1\right )}}}{x}+x = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 3.265 (sec). Leaf size: 572
DSolve[x (y'[x])^2+a y[x] y'[x]+b x==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {2 i (a+2) \log \left (\sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}-\frac {\sqrt {-a^2} y(x)}{x}\right )-2 \sqrt {-a^2} \log \left ((a+1) \left (\sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}-\frac {\sqrt {-a^2} y(x)}{x}\right )\right )-\frac {(a+2) \left (\sqrt {-a^2}+i a\right ) \log \left (\frac {\sqrt {-a^2} y(x) \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}{x}+\frac {a^2 y(x)^2}{x^2}+2 a b\right )}{a}+\frac {(a+2) \left (\sqrt {-a^2}-i a\right ) \log \left (\frac {a^3 y(x)^2}{x^2}+\frac {\sqrt {-a^2} (a+1) y(x) \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}{x}+\frac {a^2 y(x)^2}{x^2}-2 a b\right )}{a}}{8 (a+1)}=\frac {1}{2} i \log (x)+c_1,y(x)\right ] \\ \text {Solve}\left [-\frac {2 i a (a+2) \log \left (\sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}-\frac {\sqrt {-a^2} y(x)}{x}\right )+2 a \sqrt {-a^2} \log \left ((a+1) \left (\sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}-\frac {\sqrt {-a^2} y(x)}{x}\right )\right )+(a+2) \left (\sqrt {-a^2}-i a\right ) \log \left (\frac {\sqrt {-a^2} y(x) \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}{x}+\frac {a^2 y(x)^2}{x^2}+2 a b\right )-(a+2) \left (\sqrt {-a^2}+i a\right ) \log \left (\frac {a^3 y(x)^2}{x^2}+\frac {\sqrt {-a^2} (a+1) y(x) \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}{x}+\frac {a^2 y(x)^2}{x^2}-2 a b\right )}{8 a (a+1)}=c_1-\frac {1}{2} i \log (x),y(x)\right ] \\ \end{align*}