61.4.19 problem 40
Internal
problem
ID
[12124]
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
Date
solved
:
Tuesday, January 28, 2025 at 12:25:57 AM
CAS
classification
:
[_Riccati]
\begin{align*} x^{4} \left (y^{\prime }-y^{2}\right )&=a +b \,{\mathrm e}^{\frac {k}{x}}+c \,{\mathrm e}^{\frac {2 k}{x}} \end{align*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 280
dsolve(x^4*(diff(y(x),x)-y(x)^2)=a+b*exp(k/x)+c*exp(2*k/x),y(x), singsol=all)
\[
y = \frac {\left (\left (i \sqrt {a}+\frac {k}{2}\right ) \sqrt {c}-\frac {i b}{2}\right ) \operatorname {WhittakerM}\left (-\frac {i b -2 k \sqrt {c}}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right )-k c_{1} \operatorname {WhittakerW}\left (-\frac {i b -2 k \sqrt {c}}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right ) \sqrt {c}+\left (\operatorname {WhittakerW}\left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right )\right ) \left (i {\mathrm e}^{\frac {k}{x}} c +\left (-\frac {k}{2}-x \right ) \sqrt {c}+\frac {i b}{2}\right )}{\sqrt {c}\, x^{2} \left (\operatorname {WhittakerW}\left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right )\right )}
\]
✓ Solution by Mathematica
Time used: 2.079 (sec). Leaf size: 940
DSolve[x^4*(D[y[x],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 ) \operatorname {HypergeometricU}\left (\frac {\frac {i b}{\sqrt {c}}+3 k+2 i \sqrt {a}}{2 k},\frac {2 i \sqrt {a}}{k}+2,\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 ) \operatorname {HypergeometricU}\left (\frac {\frac {i b}{\sqrt {c}}+k+2 i \sqrt {a}}{2 k},\frac {2 i \sqrt {a}}{k}+1,\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 \operatorname {HypergeometricU}\left (\frac {\frac {i b}{\sqrt {c}}+k+2 i \sqrt {a}}{2 k},\frac {2 i \sqrt {a}}{k}+1,\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 ) \operatorname {HypergeometricU}\left (\frac {\frac {i b}{\sqrt {c}}+3 k+2 i \sqrt {a}}{2 k},\frac {2 i \sqrt {a}}{k}+2,\frac {2 i \sqrt {c} e^{k/x}}{k}\right )}{k \operatorname {HypergeometricU}\left (\frac {\frac {i b}{\sqrt {c}}+k+2 i \sqrt {a}}{2 k},\frac {2 i \sqrt {a}}{k}+1,\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 ) \operatorname {HypergeometricU}\left (\frac {\frac {i b}{\sqrt {c}}+3 k+2 i \sqrt {a}}{2 k},\frac {2 i \sqrt {a}}{k}+2,\frac {2 i \sqrt {c} e^{k/x}}{k}\right )}{k \operatorname {HypergeometricU}\left (\frac {\frac {i b}{\sqrt {c}}+k+2 i \sqrt {a}}{2 k},\frac {2 i \sqrt {a}}{k}+1,\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*}