Internal problem ID [10410]
Internal file name [OUTPUT/9358_Monday_June_06_2022_02_18_06_PM_56615204/index.tex]
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. Equations Containing
Exponential Functions
Problem number: 3.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "riccati"
Maple gives the following as the ode type
[_Riccati]
\[ \boxed {y^{\prime }-\sigma y^{2}=a +b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \lambda x}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \sigma \,y^{2}+a +b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \lambda x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \sigma \,y^{2}+a +b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \lambda x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=a +b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \lambda x}\), \(f_1(x)=0\) and \(f_2(x)=\sigma \). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\sigma u} \tag {1} \end {align*}
Using the above substitution in the given ODE results (after some simplification)in a second order ODE to solve for \(u(x)\) which is \begin {align*} f_2 u''(x) -\left ( f_2' + f_1 f_2 \right ) u'(x) + f_2^2 f_0 u(x) &= 0 \tag {2} \end {align*}
But \begin {align*} f_2' &=0\\ f_1 f_2 &=0\\ f_2^2 f_0 &=\sigma ^{2} \left (a +b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \lambda x}\right ) \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \sigma u^{\prime \prime }\left (x \right )+\sigma ^{2} \left (a +b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \lambda x}\right ) u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = {\mathrm e}^{-\frac {\lambda x}{2}} \left (\operatorname {WhittakerM}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{1} +\operatorname {WhittakerW}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{2} \right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {{\mathrm e}^{-\frac {\lambda x}{2}} \left (c_{1} \operatorname {WhittakerM}\left (-\frac {i b \sqrt {\sigma }-2 \lambda \sqrt {c}}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) \left (i \left (\sqrt {c}\, \sqrt {a}-\frac {b}{2}\right ) \sqrt {\sigma }+\frac {\lambda \sqrt {c}}{2}\right )-\operatorname {WhittakerW}\left (-\frac {i b \sqrt {\sigma }-2 \lambda \sqrt {c}}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{2} \lambda \sqrt {c}+\left (i c \sqrt {\sigma }\, {\mathrm e}^{\lambda x}+\frac {i b \sqrt {\sigma }}{2}-\frac {\lambda \sqrt {c}}{2}\right ) \left (\operatorname {WhittakerM}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{1} +\operatorname {WhittakerW}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{2} \right )\right )}{\sqrt {c}} \] Using the above in (1) gives the solution \[ y = -\frac {c_{1} \operatorname {WhittakerM}\left (-\frac {i b \sqrt {\sigma }-2 \lambda \sqrt {c}}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) \left (i \left (\sqrt {c}\, \sqrt {a}-\frac {b}{2}\right ) \sqrt {\sigma }+\frac {\lambda \sqrt {c}}{2}\right )-\operatorname {WhittakerW}\left (-\frac {i b \sqrt {\sigma }-2 \lambda \sqrt {c}}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{2} \lambda \sqrt {c}+\left (i c \sqrt {\sigma }\, {\mathrm e}^{\lambda x}+\frac {i b \sqrt {\sigma }}{2}-\frac {\lambda \sqrt {c}}{2}\right ) \left (\operatorname {WhittakerM}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{1} +\operatorname {WhittakerW}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{2} \right )}{\sqrt {c}\, \sigma \left (\operatorname {WhittakerM}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{1} +\operatorname {WhittakerW}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{2} \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {-c_{3} \operatorname {WhittakerM}\left (-\frac {i b \sqrt {\sigma }-2 \lambda \sqrt {c}}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) \left (i \left (\sqrt {c}\, \sqrt {a}-\frac {b}{2}\right ) \sqrt {\sigma }+\frac {\lambda \sqrt {c}}{2}\right )+\operatorname {WhittakerW}\left (-\frac {i b \sqrt {\sigma }-2 \lambda \sqrt {c}}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) \lambda \sqrt {c}+\left (-i c \sqrt {\sigma }\, {\mathrm e}^{\lambda x}-\frac {i b \sqrt {\sigma }}{2}+\frac {\lambda \sqrt {c}}{2}\right ) \left (\operatorname {WhittakerM}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{3} +\operatorname {WhittakerW}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right )\right )}{\sqrt {c}\, \sigma \left (\operatorname {WhittakerM}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{3} +\operatorname {WhittakerW}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {-c_{3} \operatorname {WhittakerM}\left (-\frac {i b \sqrt {\sigma }-2 \lambda \sqrt {c}}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) \left (i \left (\sqrt {c}\, \sqrt {a}-\frac {b}{2}\right ) \sqrt {\sigma }+\frac {\lambda \sqrt {c}}{2}\right )+\operatorname {WhittakerW}\left (-\frac {i b \sqrt {\sigma }-2 \lambda \sqrt {c}}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) \lambda \sqrt {c}+\left (-i c \sqrt {\sigma }\, {\mathrm e}^{\lambda x}-\frac {i b \sqrt {\sigma }}{2}+\frac {\lambda \sqrt {c}}{2}\right ) \left (\operatorname {WhittakerM}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{3} +\operatorname {WhittakerW}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right )\right )}{\sqrt {c}\, \sigma \left (\operatorname {WhittakerM}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{3} +\operatorname {WhittakerW}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right )\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {-c_{3} \operatorname {WhittakerM}\left (-\frac {i b \sqrt {\sigma }-2 \lambda \sqrt {c}}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) \left (i \left (\sqrt {c}\, \sqrt {a}-\frac {b}{2}\right ) \sqrt {\sigma }+\frac {\lambda \sqrt {c}}{2}\right )+\operatorname {WhittakerW}\left (-\frac {i b \sqrt {\sigma }-2 \lambda \sqrt {c}}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) \lambda \sqrt {c}+\left (-i c \sqrt {\sigma }\, {\mathrm e}^{\lambda x}-\frac {i b \sqrt {\sigma }}{2}+\frac {\lambda \sqrt {c}}{2}\right ) \left (\operatorname {WhittakerM}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{3} +\operatorname {WhittakerW}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right )\right )}{\sqrt {c}\, \sigma \left (\operatorname {WhittakerM}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{3} +\operatorname {WhittakerW}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\sigma y^{2}=a +b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \lambda x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\sigma y^{2}+a +b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \lambda x} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying Riccati trying Riccati Special trying Riccati sub-methods: trying Riccati_symmetries trying Riccati to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = -sigma*(a+b*exp(lambda*x)+c*exp(2*lambda*x))*y(x), y(x)` *** Subl Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Whittaker -> hyper3: Equivalence to 1F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 1F1 ODE <- Whittaker successful <- special function solution successful Change of variables used: [x = ln(t)/lambda] Linear ODE actually solved: (c*sigma*t^2+b*sigma*t+a*sigma)*u(t)+lambda^2*t*diff(u(t),t)+lambda^2*t^2*diff(diff(u(t),t),t) = 0 <- change of variables successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 348
dsolve(diff(y(x),x)=sigma*y(x)^2+a+b*exp(lambda*x)+c*exp(2*lambda*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {-\operatorname {WhittakerM}\left (-\frac {i \sqrt {\sigma }\, b -2 \lambda \sqrt {c}}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{x \lambda }}{\lambda }\right ) \left (i \left (\sqrt {c}\, \sqrt {a}-\frac {b}{2}\right ) \sqrt {\sigma }+\frac {\lambda \sqrt {c}}{2}\right )+\lambda c_{1} \operatorname {WhittakerW}\left (-\frac {i \sqrt {\sigma }\, b -2 \lambda \sqrt {c}}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{x \lambda }}{\lambda }\right ) \sqrt {c}+\left (-i {\mathrm e}^{x \lambda } c \sqrt {\sigma }-\frac {i \sqrt {\sigma }\, b}{2}+\frac {\lambda \sqrt {c}}{2}\right ) \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {\sigma }\, b}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{x \lambda }}{\lambda }\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {i \sqrt {\sigma }\, b}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{x \lambda }}{\lambda }\right )\right )}{\sqrt {c}\, \sigma \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {\sigma }\, b}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{x \lambda }}{\lambda }\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {i \sqrt {\sigma }\, b}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{x \lambda }}{\lambda }\right )\right )} \]
✓ Solution by Mathematica
Time used: 3.251 (sec). Leaf size: 1081
DSolve[y'[x]==sigma*y[x]^2+a+b*Exp[\[Lambda]*x]+c*Exp[2*\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {i \left (c_1 \lambda \left (\sqrt {a}-\sqrt {c} e^{\lambda x}\right ) \operatorname {HypergeometricU}\left (\frac {\frac {i \sqrt {\sigma } b}{\sqrt {c}}+\lambda +2 i \sqrt {a} \sqrt {\sigma }}{2 \lambda },\frac {2 i \sqrt {a} \sqrt {\sigma }}{\lambda }+1,\frac {2 i \sqrt {c} e^{x \lambda } \sqrt {\sigma }}{\lambda }\right )-i c_1 e^{\lambda x} \left (b \sqrt {\sigma }+\sqrt {c} \left (2 \sqrt {a} \sqrt {\sigma }-i \lambda \right )\right ) \operatorname {HypergeometricU}\left (\frac {\frac {i \sqrt {\sigma } b}{\sqrt {c}}+3 \lambda +2 i \sqrt {a} \sqrt {\sigma }}{2 \lambda },\frac {2 i \sqrt {a} \sqrt {\sigma }}{\lambda }+2,\frac {2 i \sqrt {c} e^{x \lambda } \sqrt {\sigma }}{\lambda }\right )+\lambda \left (\left (\sqrt {a}-\sqrt {c} e^{\lambda x}\right ) L_{-\frac {\frac {i \sqrt {\sigma } b}{\sqrt {c}}+\lambda +2 i \sqrt {a} \sqrt {\sigma }}{2 \lambda }}^{\frac {2 i \sqrt {a} \sqrt {\sigma }}{\lambda }}\left (\frac {2 i \sqrt {c} e^{x \lambda } \sqrt {\sigma }}{\lambda }\right )-2 \sqrt {c} e^{\lambda x} L_{-\frac {\frac {i \sqrt {\sigma } b}{\sqrt {c}}+3 \lambda +2 i \sqrt {a} \sqrt {\sigma }}{2 \lambda }}^{\frac {2 i \sqrt {a} \sqrt {\sigma }}{\lambda }+1}\left (\frac {2 i \sqrt {c} e^{x \lambda } \sqrt {\sigma }}{\lambda }\right )\right )\right )}{\lambda \sqrt {\sigma } \left (c_1 \operatorname {HypergeometricU}\left (\frac {\frac {i \sqrt {\sigma } b}{\sqrt {c}}+\lambda +2 i \sqrt {a} \sqrt {\sigma }}{2 \lambda },\frac {2 i \sqrt {a} \sqrt {\sigma }}{\lambda }+1,\frac {2 i \sqrt {c} e^{x \lambda } \sqrt {\sigma }}{\lambda }\right )+L_{-\frac {\frac {i \sqrt {\sigma } b}{\sqrt {c}}+\lambda +2 i \sqrt {a} \sqrt {\sigma }}{2 \lambda }}^{\frac {2 i \sqrt {a} \sqrt {\sigma }}{\lambda }}\left (\frac {2 i \sqrt {c} e^{x \lambda } \sqrt {\sigma }}{\lambda }\right )\right )} \\ y(x)\to \frac {-\frac {e^{\lambda x} \left (b \sqrt {\sigma }+\sqrt {c} \left (2 \sqrt {a} \sqrt {\sigma }-i \lambda \right )\right ) \operatorname {HypergeometricU}\left (\frac {\frac {i \sqrt {\sigma } b}{\sqrt {c}}+3 \lambda +2 i \sqrt {a} \sqrt {\sigma }}{2 \lambda },\frac {2 i \sqrt {a} \sqrt {\sigma }}{\lambda }+2,\frac {2 i \sqrt {c} e^{x \lambda } \sqrt {\sigma }}{\lambda }\right )}{\lambda \operatorname {HypergeometricU}\left (\frac {\frac {i \sqrt {\sigma } b}{\sqrt {c}}+\lambda +2 i \sqrt {a} \sqrt {\sigma }}{2 \lambda },\frac {2 i \sqrt {a} \sqrt {\sigma }}{\lambda }+1,\frac {2 i \sqrt {c} e^{x \lambda } \sqrt {\sigma }}{\lambda }\right )}-i \left (\sqrt {a}-\sqrt {c} e^{\lambda x}\right )}{\sqrt {\sigma }} \\ y(x)\to \frac {-\frac {e^{\lambda x} \left (b \sqrt {\sigma }+\sqrt {c} \left (2 \sqrt {a} \sqrt {\sigma }-i \lambda \right )\right ) \operatorname {HypergeometricU}\left (\frac {\frac {i \sqrt {\sigma } b}{\sqrt {c}}+3 \lambda +2 i \sqrt {a} \sqrt {\sigma }}{2 \lambda },\frac {2 i \sqrt {a} \sqrt {\sigma }}{\lambda }+2,\frac {2 i \sqrt {c} e^{x \lambda } \sqrt {\sigma }}{\lambda }\right )}{\lambda \operatorname {HypergeometricU}\left (\frac {\frac {i \sqrt {\sigma } b}{\sqrt {c}}+\lambda +2 i \sqrt {a} \sqrt {\sigma }}{2 \lambda },\frac {2 i \sqrt {a} \sqrt {\sigma }}{\lambda }+1,\frac {2 i \sqrt {c} e^{x \lambda } \sqrt {\sigma }}{\lambda }\right )}-i \left (\sqrt {a}-\sqrt {c} e^{\lambda x}\right )}{\sqrt {\sigma }} \\ \end{align*}