Internal problem ID [4109]
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]
\[ \boxed {{y^{\prime }}^{2} x +a y y^{\prime }=-b x} \]
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 217
dsolve(x*diff(y(x),x)^2+a*y(x)*diff(y(x),x)+b*x = 0,y(x), singsol=all)
\begin{align*} \frac {-c_{1} 2^{\frac {2+a}{2+2 a}} \left (a y \left (x \right )-\sqrt {a^{2} y \left (x \right )^{2}-4 b \,x^{2}}\right ) {\left (\frac {a \left (-y \left (x \right ) \left (a +1\right ) \sqrt {a^{2} y \left (x \right )^{2}-4 b \,x^{2}}+\left (a^{2}+a \right ) y \left (x \right )^{2}-2 b \,x^{2}\right )}{x^{2}}\right )}^{\frac {-a -2}{2+2 a}}+x^{2}}{x} &= 0 \\ \frac {c_{1} 2^{\frac {2+a}{2+2 a}} \left (a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b \,x^{2}}\right ) {\left (\frac {a \left (y \left (x \right ) \left (a +1\right ) \sqrt {a^{2} y \left (x \right )^{2}-4 b \,x^{2}}+\left (a^{2}+a \right ) y \left (x \right )^{2}-2 b \,x^{2}\right )}{x^{2}}\right )}^{\frac {-a -2}{2+2 a}}+x^{2}}{x} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.082 (sec). Leaf size: 423
DSolve[x (y'[x])^2+a y[x] y'[x]+b x==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [-\frac {i \left (2 \log \left (-i \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {a y(x)}{x}+2 i \sqrt {b}\right )+2 (a+1) \log \left (i \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {a y(x)}{x}-2 i \sqrt {b}\right )-(a+2) \log \left (\frac {i (a+2) y(x) \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}{x}+2 \sqrt {b} \left (\sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}-\frac {i (a+2) y(x)}{x}\right )+\frac {a^2 y(x)^2}{x^2}-4 b\right )\right )}{4 (a+1)}&=c_1-\frac {1}{2} i \log (x),y(x)\right ] \\ \text {Solve}\left [\frac {i \left (2 (a+1) \log \left (-i \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {a y(x)}{x}+2 i \sqrt {b}\right )+2 \log \left (i \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {a y(x)}{x}-2 i \sqrt {b}\right )-(a+2) \log \left (-\frac {i (a+2) y(x) \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}{x}+2 \sqrt {b} \left (\sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {i (a+2) y(x)}{x}\right )+\frac {a^2 y(x)^2}{x^2}-4 b\right )\right )}{4 (a+1)}&=\frac {1}{2} i \log (x)+c_1,y(x)\right ] \\ \end{align*}