Internal problem ID [3321]
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]
\[ \boxed {y^{\prime }-c y^{2}=a \,x^{n -1}+b \,x^{2 n}} \]
✓ Solution by Maple
Time used: 0.0 (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)
\[ y \left (x \right ) = -\frac {\left (-2 b^{\frac {3}{2}} c_{1} n -2 b^{\frac {3}{2}} c_{1} \right ) \operatorname {WhittakerW}\left (-\frac {i \sqrt {c}\, a -2 \sqrt {b}\, n -2 \sqrt {b}}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right )}{2 b^{\frac {3}{2}} \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {c}\, a}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {i \sqrt {c}\, a}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right )\right ) c x}-\frac {\left (2 i x^{n +1} \sqrt {c}\, c_{1} b^{2}+i \sqrt {c}\, c_{1} a b -b^{\frac {3}{2}} c_{1} n \right ) \operatorname {WhittakerW}\left (-\frac {i \sqrt {c}\, a}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right )+\left (-i \sqrt {c}\, a b +b^{\frac {3}{2}} n +2 b^{\frac {3}{2}}\right ) \operatorname {WhittakerM}\left (-\frac {i \sqrt {c}\, a -2 \sqrt {b}\, n -2 \sqrt {b}}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right )+\left (2 i x^{n +1} \sqrt {c}\, b^{2}+i \sqrt {c}\, a b -b^{\frac {3}{2}} n \right ) \operatorname {WhittakerM}\left (-\frac {i \sqrt {c}\, a}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right )}{2 b^{\frac {3}{2}} \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {c}\, a}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {i \sqrt {c}\, a}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {c}\, \sqrt {b}\, x^{n +1}}{n +1}\right )\right ) c x} \]
✓ Solution by Mathematica
Time used: 1.115 (sec). Leaf size: 982
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} \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} a}{\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 ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} a}{\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 (L_{-\frac {\sqrt {c} a}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+2 L_{-\frac {\sqrt {c} a}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {3 n+2}{2 n+2}}^{\frac {n}{n+1}}\left (\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )\right )}{\sqrt {c} (n+1)^2 \left (L_{-\frac {\sqrt {c} a}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} a}{\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 ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} a}{\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 )}{\operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} a}{\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} y(x)\to \frac {x^n \left (-\frac {\left (a \sqrt {c} (n+1)+\sqrt {b} \sqrt {-(n+1)^2} n\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} a}{\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 )}{\operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} a}{\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*}