Internal problem ID [2812]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 3
Problem number: 57.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-a \,x^{n -1}-b \,x^{2 n}-y^{2} c=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 499
dsolve(diff(y(x),x) = a*x^(n-1)+b*x^(2*n)+c*y(x)^2,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (-2 b^{\frac {3}{2}} c_{1} n -2 b^{\frac {3}{2}} c_{1}\right ) \WhittakerW \left (-\frac {i \sqrt {c}\, a -2 \sqrt {b}\, n -2 \sqrt {b}}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right )}{2 b^{\frac {3}{2}} \left (\WhittakerW \left (-\frac {i \sqrt {c}\, a}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right ) c_{1}+\WhittakerM \left (-\frac {i \sqrt {c}\, a}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right )\right ) c x}-\frac {\left (2 i \sqrt {c}\, x^{n +1} c_{1} b^{2}+i \sqrt {c}\, c_{1} a b -b^{\frac {3}{2}} c_{1} n \right ) \WhittakerW \left (-\frac {i \sqrt {c}\, a}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right )+\left (-i \sqrt {c}\, a b +b^{\frac {3}{2}} n +2 b^{\frac {3}{2}}\right ) \WhittakerM \left (-\frac {i \sqrt {c}\, a -2 \sqrt {b}\, n -2 \sqrt {b}}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right )+\left (2 i \sqrt {c}\, x^{n +1} b^{2}+i \sqrt {c}\, a b -b^{\frac {3}{2}} n \right ) \WhittakerM \left (-\frac {i \sqrt {c}\, a}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right )}{2 b^{\frac {3}{2}} \left (\WhittakerW \left (-\frac {i \sqrt {c}\, a}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right ) c_{1}+\WhittakerM \left (-\frac {i \sqrt {c}\, a}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right )\right ) c x} \]
✓ Solution by Mathematica
Time used: 0.944 (sec). Leaf size: 764
DSolve[y'[x]==a x^(n-1)+b x^(2 n)+c y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x^n \left (\sqrt {b} c_1 (n+1) \sqrt {-(n+1)^2} \text {HypergeometricU}\left (\frac {1}{2} \left (\frac {a \sqrt {c}}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_1 \left (a \sqrt {c} (n+1)+\sqrt {b} \sqrt {-(n+1)^2} n\right ) \text {HypergeometricU}\left (\frac {1}{2} \left (\frac {a \sqrt {c}}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {3 n+2}{n+1}\right ),\frac {n}{n+1}+1,\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+\sqrt {b} (n+1) \sqrt {-(n+1)^2} \left (\text {LaguerreL}\left (-\frac {a \sqrt {c}}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)},-\frac {1}{n+1},\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+2 \text {LaguerreL}\left (-\frac {a \sqrt {c}}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {3 n+2}{2 n+2},\frac {n}{n+1},\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )\right )}{\sqrt {c} (n+1)^2 \left (\text {LaguerreL}\left (-\frac {a \sqrt {c}}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)},-\frac {1}{n+1},\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_1 \text {HypergeometricU}\left (\frac {1}{2} \left (\frac {a \sqrt {c}}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )} \\ y(x)\to \frac {x^n \left (-\frac {\left (a \sqrt {c} (n+1)+\sqrt {b} \sqrt {-(n+1)^2} n\right ) \text {HypergeometricU}\left (\frac {1}{2} \left (\frac {a \sqrt {c}}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}+2\right ),\frac {n}{n+1}+1,\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )}{\text {HypergeometricU}\left (\frac {1}{2} \left (\frac {a \sqrt {c}}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )}-\sqrt {b} \sqrt {-(n+1)^2} (n+1)\right )}{\sqrt {c} (n+1)^2} \\ \end{align*}