Internal problem ID [9705]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.3-2. Equations with power and
exponential functions
Problem number: 40.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {x^{4} \left (y^{\prime }-y^{2}\right )-a -b \,{\mathrm e}^{\frac {k}{x}}-c \,{\mathrm e}^{\frac {2 k}{x}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 564
dsolve(x^4*(diff(y(x),x)-y(x)^2)=a+b*exp(k/x)+c*exp(2*k/x),y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (2 i \WhittakerW \left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right ) c_{1} c^{2}+2 i \WhittakerM \left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right ) c^{2}\right ) {\mathrm e}^{\frac {k}{x}}}{2 c^{\frac {3}{2}} x^{2} \left (\WhittakerW \left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right ) c_{1}+\WhittakerM \left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right )\right )}-\frac {c_{1} k \WhittakerW \left (-\frac {i b -2 k \sqrt {c}}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right )}{x^{2} \left (\WhittakerW \left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right ) c_{1}+\WhittakerM \left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right )\right )}+\frac {\left (-c^{\frac {3}{2}} c_{1} k -2 c^{\frac {3}{2}} c_{1} x +i c_{1} b c \right ) \WhittakerW \left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right )+\left (2 i \sqrt {a}\, c^{\frac {3}{2}}+c^{\frac {3}{2}} k -i b c \right ) \WhittakerM \left (-\frac {i b -2 k \sqrt {c}}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right )+\left (-c^{\frac {3}{2}} k -2 c^{\frac {3}{2}} x +i b c \right ) \WhittakerM \left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right )}{2 c^{\frac {3}{2}} x^{2} \left (\WhittakerW \left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right ) c_{1}+\WhittakerM \left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right )\right )} \]
✓ Solution by Mathematica
Time used: 1.683 (sec). Leaf size: 940
DSolve[x^4*(y'[x]-y[x]^2)==a+b*Exp[k/x]+c*Exp[2*k/x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{k/x} \log \left (e^{k/x}\right ) \left (c_1 \left (b+\sqrt {c} \left (2 \sqrt {a}-i k\right )\right ) \text {HypergeometricU}\left (\frac {2 i \sqrt {a}+\frac {i b}{\sqrt {c}}+3 k}{2 k},2+\frac {2 i \sqrt {a}}{k},\frac {2 i \sqrt {c} e^{k/x}}{k}\right )-2 i \sqrt {c} k L_{-\frac {\frac {i b}{\sqrt {c}}+3 k+2 i \sqrt {a}}{2 k}}^{\frac {2 i \sqrt {a}}{k}+1}\left (\frac {2 i \sqrt {c} e^{k/x}}{k}\right )\right )-c_1 k \left (k-i \log \left (e^{k/x}\right ) \left (\sqrt {a}-\sqrt {c} e^{k/x}\right )\right ) \text {HypergeometricU}\left (\frac {2 i \sqrt {a}+\frac {i b}{\sqrt {c}}+k}{2 k},1+\frac {2 i \sqrt {a}}{k},\frac {2 i \sqrt {c} e^{k/x}}{k}\right )-k \left (k-i \log \left (e^{k/x}\right ) \left (\sqrt {a}-\sqrt {c} e^{k/x}\right )\right ) L_{-\frac {\frac {i b}{\sqrt {c}}+k+2 i \sqrt {a}}{2 k}}^{\frac {2 i \sqrt {a}}{k}}\left (\frac {2 i \sqrt {c} e^{k/x}}{k}\right )}{k x^2 \log \left (e^{k/x}\right ) \left (c_1 \text {HypergeometricU}\left (\frac {2 i \sqrt {a}+\frac {i b}{\sqrt {c}}+k}{2 k},1+\frac {2 i \sqrt {a}}{k},\frac {2 i \sqrt {c} e^{k/x}}{k}\right )+L_{-\frac {\frac {i b}{\sqrt {c}}+k+2 i \sqrt {a}}{2 k}}^{\frac {2 i \sqrt {a}}{k}}\left (\frac {2 i \sqrt {c} e^{k/x}}{k}\right )\right )} \\ y(x)\to \frac {\frac {e^{k/x} \left (b+\sqrt {c} \left (2 \sqrt {a}-i k\right )\right ) \text {HypergeometricU}\left (\frac {2 i \sqrt {a}+\frac {i b}{\sqrt {c}}+3 k}{2 k},2+\frac {2 i \sqrt {a}}{k},\frac {2 i \sqrt {c} e^{k/x}}{k}\right )}{k \text {HypergeometricU}\left (\frac {2 i \sqrt {a}+\frac {i b}{\sqrt {c}}+k}{2 k},1+\frac {2 i \sqrt {a}}{k},\frac {2 i \sqrt {c} e^{k/x}}{k}\right )}+i \left (\sqrt {a}-\sqrt {c} e^{k/x}\right )-\frac {k}{\log \left (e^{k/x}\right )}}{x^2} \\ y(x)\to \frac {\frac {e^{k/x} \left (b+\sqrt {c} \left (2 \sqrt {a}-i k\right )\right ) \text {HypergeometricU}\left (\frac {2 i \sqrt {a}+\frac {i b}{\sqrt {c}}+3 k}{2 k},2+\frac {2 i \sqrt {a}}{k},\frac {2 i \sqrt {c} e^{k/x}}{k}\right )}{k \text {HypergeometricU}\left (\frac {2 i \sqrt {a}+\frac {i b}{\sqrt {c}}+k}{2 k},1+\frac {2 i \sqrt {a}}{k},\frac {2 i \sqrt {c} e^{k/x}}{k}\right )}+i \left (\sqrt {a}-\sqrt {c} e^{k/x}\right )-\frac {k}{\log \left (e^{k/x}\right )}}{x^2} \\ \end{align*}