Internal problem ID [7606]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 25.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }+a y^{2}-b \,x^{2 \nu }-c \,x^{\nu -1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 378
dsolve(diff(y(x),x) + a*y(x)^2 - b*x^(2*nu) - c*x^(nu-1)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (2 \sqrt {a}\, x^{\nu +1} c_{1} b^{2}-b^{\frac {3}{2}} c_{1} \nu +\sqrt {a}\, c_{1} b c \right ) \WhittakerW \left (-\frac {\sqrt {a}\, c}{2 \sqrt {b}\, \left (\nu +1\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right )+\left (-2 b^{\frac {3}{2}} c_{1} \nu -2 b^{\frac {3}{2}} c_{1}\right ) \WhittakerW \left (-\frac {\sqrt {a}\, c -2 \sqrt {b}\, \nu -2 \sqrt {b}}{2 \sqrt {b}\, \left (\nu +1\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right )+\left (2 \sqrt {a}\, x^{\nu +1} b^{2}-b^{\frac {3}{2}} \nu +\sqrt {a}\, b c \right ) \WhittakerM \left (-\frac {\sqrt {a}\, c}{2 \sqrt {b}\, \left (\nu +1\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right )+\left (b^{\frac {3}{2}} \nu -\sqrt {a}\, b c +2 b^{\frac {3}{2}}\right ) \WhittakerM \left (-\frac {\sqrt {a}\, c -2 \sqrt {b}\, \nu -2 \sqrt {b}}{2 \sqrt {b}\, \left (\nu +1\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right )}{2 b^{\frac {3}{2}} \left (\WhittakerW \left (-\frac {\sqrt {a}\, c}{2 \sqrt {b}\, \left (\nu +1\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right ) c_{1}+\WhittakerM \left (-\frac {\sqrt {a}\, c}{2 \sqrt {b}\, \left (\nu +1\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right )\right ) a x} \]
✓ Solution by Mathematica
Time used: 1.559 (sec). Leaf size: 928
DSolve[y'[x] + a*y[x]^2 - b*x^(2*nu) - c*x^(nu-1)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x^{\nu } \left (\sqrt {b} c_1 (\nu +1) \sqrt {(\nu +1)^2} \text {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {(\nu +1)^2}}+\frac {\nu }{\nu +1}\right ),\frac {\nu }{\nu +1},\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {(\nu +1)^2}}\right )+c_1 \left (\sqrt {a} c (\nu +1)+\sqrt {b} \sqrt {(\nu +1)^2} \nu \right ) \text {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {(\nu +1)^2}}+\frac {3 \nu +2}{\nu +1}\right ),\frac {\nu }{\nu +1}+1,\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {(\nu +1)^2}}\right )+\sqrt {b} (\nu +1) \sqrt {(\nu +1)^2} \left (L_{-\frac {\sqrt {a} c}{2 \sqrt {b} \sqrt {(\nu +1)^2}}-\frac {\nu }{2 (\nu +1)}}^{-\frac {1}{\nu +1}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {(\nu +1)^2}}\right )+2 L_{-\frac {\sqrt {a} c}{2 \sqrt {b} \sqrt {(\nu +1)^2}}-\frac {3 \nu +2}{2 \nu +2}}^{\frac {\nu }{\nu +1}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {(\nu +1)^2}}\right )\right )\right )}{\sqrt {a} (\nu +1)^2 \left (L_{-\frac {\sqrt {a} c}{2 \sqrt {b} \sqrt {(\nu +1)^2}}-\frac {\nu }{2 (\nu +1)}}^{-\frac {1}{\nu +1}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {(\nu +1)^2}}\right )+c_1 \text {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {(\nu +1)^2}}+\frac {\nu }{\nu +1}\right ),\frac {\nu }{\nu +1},\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {(\nu +1)^2}}\right )\right )} \\ y(x)\to \frac {x^{\nu } \left (-\frac {\left (\sqrt {a} c (\nu +1)+\sqrt {b} \sqrt {(\nu +1)^2} \nu \right ) \text {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {(\nu +1)^2}}+\frac {\nu }{\nu +1}+2\right ),\frac {\nu }{\nu +1}+1,\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {(\nu +1)^2}}\right )}{\text {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {(\nu +1)^2}}+\frac {\nu }{\nu +1}\right ),\frac {\nu }{\nu +1},\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {(\nu +1)^2}}\right )}-\sqrt {b} \sqrt {(\nu +1)^2} (\nu +1)\right )}{\sqrt {a} (\nu +1)^2} \\ y(x)\to \frac {x^{\nu } \left (-\frac {\left (\sqrt {a} c (\nu +1)+\sqrt {b} \sqrt {(\nu +1)^2} \nu \right ) \text {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {(\nu +1)^2}}+\frac {\nu }{\nu +1}+2\right ),\frac {\nu }{\nu +1}+1,\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {(\nu +1)^2}}\right )}{\text {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {(\nu +1)^2}}+\frac {\nu }{\nu +1}\right ),\frac {\nu }{\nu +1},\frac {2 \sqrt {a} \sqrt {b} x^{\nu +1}}{\sqrt {(\nu +1)^2}}\right )}-\sqrt {b} \sqrt {(\nu +1)^2} (\nu +1)\right )}{\sqrt {a} (\nu +1)^2} \\ \end{align*}