Internal problem ID [10414]
Internal file name [OUTPUT/9362_Monday_June_06_2022_02_18_15_PM_89074729/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: 7.
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]
\[ \boxed {y^{\prime }-y^{2}=a \,{\mathrm e}^{2 \lambda x} \left ({\mathrm e}^{\lambda x}+b \right )^{n}-\frac {\lambda ^{2}}{4}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= y^{2}+a \,{\mathrm e}^{2 \lambda x} \left ({\mathrm e}^{\lambda x}+b \right )^{n}-\frac {\lambda ^{2}}{4} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = y^{2}+a \,{\mathrm e}^{2 \lambda x} \left ({\mathrm e}^{\lambda x}+b \right )^{n}-\frac {\lambda ^{2}}{4} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=a \,{\mathrm e}^{2 \lambda x} \left ({\mathrm e}^{\lambda x}+b \right )^{n}-\frac {\lambda ^{2}}{4}\), \(f_1(x)=0\) and \(f_2(x)=1\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{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 &=a \,{\mathrm e}^{2 \lambda x} \left ({\mathrm e}^{\lambda x}+b \right )^{n}-\frac {\lambda ^{2}}{4} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} u^{\prime \prime }\left (x \right )+\left (a \,{\mathrm e}^{2 \lambda x} \left ({\mathrm e}^{\lambda x}+b \right )^{n}-\frac {\lambda ^{2}}{4}\right ) u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \frac {-\csc \left (\frac {\pi \left (n +3\right )}{2+n}\right ) c_{1} \operatorname {BesselI}\left (-\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) \pi {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {1}{4+2 n}} {\mathrm e}^{-\frac {\lambda x}{2}}+\Gamma \left (\frac {n +3}{2+n}\right )^{2} {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{-\frac {1}{4+2 n}} \operatorname {BesselI}\left (\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) c_{2} \left (2+n \right ) \left ({\mathrm e}^{\frac {\lambda x}{2}}+{\mathrm e}^{-\frac {\lambda x}{2}} b \right )}{\left (2+n \right ) \Gamma \left (\frac {n +3}{2+n}\right )} \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\lambda \left (\Gamma \left (\frac {n +3}{2+n}\right )^{2} {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {1+n}{4+2 n}} c_{2} \left (2+n \right )^{2} \left ({\mathrm e}^{\frac {\lambda x}{2}} b +{\mathrm e}^{\frac {3 \lambda x}{2}}\right ) \operatorname {BesselI}\left (\frac {n +3}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right )-\frac {\Gamma \left (\frac {n +3}{2+n}\right )^{2} {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{-\frac {1}{4+2 n}} \operatorname {BesselI}\left (\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) c_{2} \left (2+n \right ) \left ({\mathrm e}^{-\frac {\lambda x}{2}} b^{2}-{\mathrm e}^{\frac {3 \lambda x}{2}}\right )}{2}+\frac {\left (\left ({\mathrm e}^{\frac {\lambda x}{2}}+{\mathrm e}^{-\frac {\lambda x}{2}} b \right ) \operatorname {BesselI}\left (-\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {1}{4+2 n}}-2 \,{\mathrm e}^{\frac {\lambda x}{2}} \operatorname {BesselI}\left (\frac {1+n}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {n +3}{4+2 n}} \left (2+n \right )\right ) \csc \left (\frac {\pi \left (n +3\right )}{2+n}\right ) \pi c_{1}}{2}\right )}{\left ({\mathrm e}^{\lambda x}+b \right ) \left (2+n \right ) \Gamma \left (\frac {n +3}{2+n}\right )} \] Using the above in (1) gives the solution \[ y = -\frac {\lambda \left (\Gamma \left (\frac {n +3}{2+n}\right )^{2} {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {1+n}{4+2 n}} c_{2} \left (2+n \right )^{2} \left ({\mathrm e}^{\frac {\lambda x}{2}} b +{\mathrm e}^{\frac {3 \lambda x}{2}}\right ) \operatorname {BesselI}\left (\frac {n +3}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right )-\frac {\Gamma \left (\frac {n +3}{2+n}\right )^{2} {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{-\frac {1}{4+2 n}} \operatorname {BesselI}\left (\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) c_{2} \left (2+n \right ) \left ({\mathrm e}^{-\frac {\lambda x}{2}} b^{2}-{\mathrm e}^{\frac {3 \lambda x}{2}}\right )}{2}+\frac {\left (\left ({\mathrm e}^{\frac {\lambda x}{2}}+{\mathrm e}^{-\frac {\lambda x}{2}} b \right ) \operatorname {BesselI}\left (-\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {1}{4+2 n}}-2 \,{\mathrm e}^{\frac {\lambda x}{2}} \operatorname {BesselI}\left (\frac {1+n}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {n +3}{4+2 n}} \left (2+n \right )\right ) \csc \left (\frac {\pi \left (n +3\right )}{2+n}\right ) \pi c_{1}}{2}\right )}{\left ({\mathrm e}^{\lambda x}+b \right ) \left (-\csc \left (\frac {\pi \left (n +3\right )}{2+n}\right ) c_{1} \operatorname {BesselI}\left (-\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) \pi {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {1}{4+2 n}} {\mathrm e}^{-\frac {\lambda x}{2}}+\Gamma \left (\frac {n +3}{2+n}\right )^{2} {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{-\frac {1}{4+2 n}} \operatorname {BesselI}\left (\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) c_{2} \left (2+n \right ) \left ({\mathrm e}^{\frac {\lambda x}{2}}+{\mathrm e}^{-\frac {\lambda x}{2}} b \right )\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 {\lambda \left (-2 \Gamma \left (\frac {n +3}{2+n}\right )^{2} {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {1+n}{4+2 n}} \left (2+n \right )^{2} \left (b \,{\mathrm e}^{\lambda x}+{\mathrm e}^{2 \lambda x}\right ) \operatorname {BesselI}\left (\frac {n +3}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right )+\Gamma \left (\frac {n +3}{2+n}\right )^{2} {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{-\frac {1}{4+2 n}} \operatorname {BesselI}\left (\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) \left (b^{2}-{\mathrm e}^{2 \lambda x}\right ) \left (2+n \right )-\csc \left (\frac {\pi \left (n +3\right )}{2+n}\right ) \pi \left (\left ({\mathrm e}^{\lambda x}+b \right ) \operatorname {BesselI}\left (-\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {1}{4+2 n}}-2 \,{\mathrm e}^{\lambda x} \operatorname {BesselI}\left (\frac {1+n}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) \left (2+n \right ) {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {n +3}{4+2 n}}\right ) c_{3} \right )}{2 \left ({\mathrm e}^{\lambda x}+b \right ) \left (-c_{3} \csc \left (\frac {\pi \left (n +3\right )}{2+n}\right ) \pi \operatorname {BesselI}\left (-\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {1}{4+2 n}}+\Gamma \left (\frac {n +3}{2+n}\right )^{2} {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{-\frac {1}{4+2 n}} \operatorname {BesselI}\left (\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) \left (2+n \right ) \left ({\mathrm e}^{\lambda x}+b \right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\lambda \left (-2 \Gamma \left (\frac {n +3}{2+n}\right )^{2} {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {1+n}{4+2 n}} \left (2+n \right )^{2} \left (b \,{\mathrm e}^{\lambda x}+{\mathrm e}^{2 \lambda x}\right ) \operatorname {BesselI}\left (\frac {n +3}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right )+\Gamma \left (\frac {n +3}{2+n}\right )^{2} {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{-\frac {1}{4+2 n}} \operatorname {BesselI}\left (\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) \left (b^{2}-{\mathrm e}^{2 \lambda x}\right ) \left (2+n \right )-\csc \left (\frac {\pi \left (n +3\right )}{2+n}\right ) \pi \left (\left ({\mathrm e}^{\lambda x}+b \right ) \operatorname {BesselI}\left (-\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {1}{4+2 n}}-2 \,{\mathrm e}^{\lambda x} \operatorname {BesselI}\left (\frac {1+n}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) \left (2+n \right ) {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {n +3}{4+2 n}}\right ) c_{3} \right )}{2 \left ({\mathrm e}^{\lambda x}+b \right ) \left (-c_{3} \csc \left (\frac {\pi \left (n +3\right )}{2+n}\right ) \pi \operatorname {BesselI}\left (-\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {1}{4+2 n}}+\Gamma \left (\frac {n +3}{2+n}\right )^{2} {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{-\frac {1}{4+2 n}} \operatorname {BesselI}\left (\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) \left (2+n \right ) \left ({\mathrm e}^{\lambda x}+b \right )\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\lambda \left (-2 \Gamma \left (\frac {n +3}{2+n}\right )^{2} {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {1+n}{4+2 n}} \left (2+n \right )^{2} \left (b \,{\mathrm e}^{\lambda x}+{\mathrm e}^{2 \lambda x}\right ) \operatorname {BesselI}\left (\frac {n +3}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right )+\Gamma \left (\frac {n +3}{2+n}\right )^{2} {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{-\frac {1}{4+2 n}} \operatorname {BesselI}\left (\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) \left (b^{2}-{\mathrm e}^{2 \lambda x}\right ) \left (2+n \right )-\csc \left (\frac {\pi \left (n +3\right )}{2+n}\right ) \pi \left (\left ({\mathrm e}^{\lambda x}+b \right ) \operatorname {BesselI}\left (-\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {1}{4+2 n}}-2 \,{\mathrm e}^{\lambda x} \operatorname {BesselI}\left (\frac {1+n}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) \left (2+n \right ) {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {n +3}{4+2 n}}\right ) c_{3} \right )}{2 \left ({\mathrm e}^{\lambda x}+b \right ) \left (-c_{3} \csc \left (\frac {\pi \left (n +3\right )}{2+n}\right ) \pi \operatorname {BesselI}\left (-\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {1}{4+2 n}}+\Gamma \left (\frac {n +3}{2+n}\right )^{2} {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{-\frac {1}{4+2 n}} \operatorname {BesselI}\left (\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) \left (2+n \right ) \left ({\mathrm e}^{\lambda x}+b \right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2}=a \,{\mathrm e}^{2 \lambda x} \left ({\mathrm e}^{\lambda x}+b \right )^{n}-\frac {\lambda ^{2}}{4} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2}+a \,{\mathrm e}^{2 \lambda x} \left ({\mathrm e}^{\lambda x}+b \right )^{n}-\frac {\lambda ^{2}}{4} \end {array} \]
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) = (-a*exp(2*lambda*x)*(exp(lambda*x)+b)^n+(1/4)*lambda^2)*y(x), y(x)` 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 symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying an equivalence, under non-integer power transformations, to LODEs admitting Liouvillian solutions. -> 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 0F1 ODE <- Whittaker successful <- special function solution successful Change of variables used: [x = ln(t)/lambda] Linear ODE actually solved: (4*a*t^2*(t+b)^n-lambda^2)*u(t)+4*lambda^2*t*diff(u(t),t)+4*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: 1342
dsolve(diff(y(x),x)=y(x)^2+a*exp(2*lambda*x)*(exp(lambda*x)+b)^n-1/4*lambda^2,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==y[x]^2+a*Exp[2*\[Lambda]*x]*(Exp[\[Lambda]*x]+b)^n-1/4*\[Lambda]^2,y[x],x,IncludeSingularSolutions -> True]
Not solved