Internal problem ID [8812]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 477.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class C`], _rational, _dAlembert]
\[ \boxed {a y {y^{\prime }}^{2}+\left (2 x -b \right ) y^{\prime }-y=0} \]
✓ Solution by Maple
Time used: 0.312 (sec). Leaf size: 741
dsolve(a*y(x)*diff(y(x),x)^2+(2*x-b)*diff(y(x),x)-y(x) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {-2 x +b}{2 \sqrt {-a}} \\ y \left (x \right ) &= \frac {-2 x +b}{2 \sqrt {-a}} \\ y \left (x \right ) &= 0 \\ -4 a \left (\int _{}^{y \left (x \right )}\frac {16 \textit {\_f} \left (\frac {1}{16}+\left (\left (\frac {b}{4}-\frac {x}{2}\right ) \sqrt {4 a \,\textit {\_f}^{2}+b^{2}-4 b x +4 x^{2}}+a \,\textit {\_f}^{2}+\frac {b^{2}}{4}-b x +x^{2}\right ) \left (\int _{\textit {\_b}}^{x}\frac {-2 a \,\textit {\_f}^{2}+2 \sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}}\, \textit {\_a} -\sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}}\, b -4 \textit {\_a}^{2}+4 \textit {\_a} b -b^{2}}{\left (-4 a \,\textit {\_f}^{2}+2 \sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}}\, \textit {\_a} -\sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}}\, b -4 \textit {\_a}^{2}+4 \textit {\_a} b -b^{2}\right )^{2} \sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}}}d \textit {\_a} \right )\right )}{\left (-2 x +b \right ) \sqrt {4 a \,\textit {\_f}^{2}+b^{2}-4 b x +4 x^{2}}+4 a \,\textit {\_f}^{2}+b^{2}-4 b x +4 x^{2}}d \textit {\_f} \right )+2 \left (\int _{\textit {\_b}}^{x}\frac {-2 \textit {\_a} +b +\sqrt {4 a y \left (x \right )^{2}+\left (-2 \textit {\_a} +b \right )^{2}}}{\left (-2 \textit {\_a} +b \right ) \sqrt {4 a y \left (x \right )^{2}+\left (-2 \textit {\_a} +b \right )^{2}}+4 a y \left (x \right )^{2}+\left (-2 \textit {\_a} +b \right )^{2}}d \textit {\_a} \right )+c_{1} &= 0 \\ -4 a \left (\int _{}^{y \left (x \right )}\frac {16 \textit {\_f} \left (\frac {1}{16}+\left (\left (-\frac {b}{4}+\frac {x}{2}\right ) \sqrt {4 a \,\textit {\_f}^{2}+b^{2}-4 b x +4 x^{2}}+a \,\textit {\_f}^{2}+\frac {b^{2}}{4}-b x +x^{2}\right ) \left (\int _{\textit {\_b}}^{x}\frac {2 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}+2 \sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}}\, \textit {\_a} -\sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}}\, b}{\left (4 a \,\textit {\_f}^{2}+2 \sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}}\, \textit {\_a} -\sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}}\, b +4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}\right )^{2} \sqrt {4 a \,\textit {\_f}^{2}+4 \textit {\_a}^{2}-4 \textit {\_a} b +b^{2}}}d \textit {\_a} \right )\right )}{\left (2 x -b \right ) \sqrt {4 a \,\textit {\_f}^{2}+b^{2}-4 b x +4 x^{2}}+4 a \,\textit {\_f}^{2}+b^{2}-4 b x +4 x^{2}}d \textit {\_f} \right )-2 \left (\int _{\textit {\_b}}^{x}\frac {2 \textit {\_a} -b +\sqrt {4 a y \left (x \right )^{2}+\left (-2 \textit {\_a} +b \right )^{2}}}{\left (2 \textit {\_a} -b \right ) \sqrt {4 a y \left (x \right )^{2}+\left (-2 \textit {\_a} +b \right )^{2}}+4 a y \left (x \right )^{2}+\left (-2 \textit {\_a} +b \right )^{2}}d \textit {\_a} \right )+c_{1} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.873 (sec). Leaf size: 187
DSolve[-y[x] + (-b + 2*x)*y'[x] + a*y[x]*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {2} e^{\frac {c_1}{2}} \sqrt {2 a e^{c_1}+b-2 x} \\ y(x)\to \sqrt {2} e^{\frac {c_1}{2}} \sqrt {2 a e^{c_1}+b-2 x} \\ y(x)\to -\frac {e^{\frac {c_1}{2}} \sqrt {-2 b+4 x+e^{c_1}}}{2 \sqrt {a}} \\ y(x)\to \frac {e^{\frac {c_1}{2}} \sqrt {-2 b+4 x+e^{c_1}}}{2 \sqrt {a}} \\ y(x)\to 0 \\ y(x)\to -\frac {i (b-2 x)}{2 \sqrt {a}} \\ y(x)\to \frac {i (b-2 x)}{2 \sqrt {a}} \\ \end{align*}