Internal problem ID [10437]
Internal file name [OUTPUT/9385_Monday_June_06_2022_02_21_05_PM_40185526/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 1.2.3-2. Equations with power and
exponential functions
Problem number: 30.
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]
\[ \boxed {y^{\prime }-a \,x^{n} y^{2}+a b \,x^{n} {\mathrm e}^{\lambda x} y=b \lambda \,{\mathrm e}^{\lambda x}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= a \,x^{n} y^{2}-a b \,x^{n} {\mathrm e}^{\lambda x} y +b \lambda \,{\mathrm e}^{\lambda x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = a \,x^{n} y^{2}-a b \,x^{n} {\mathrm e}^{\lambda x} y +b \lambda \,{\mathrm e}^{\lambda x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=b \lambda \,{\mathrm e}^{\lambda x}\), \(f_1(x)=-a b \,x^{n} {\mathrm e}^{\lambda x}\) and \(f_2(x)=x^{n} a\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{x^{n} a 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' &=\frac {x^{n} n a}{x}\\ f_1 f_2 &=-a^{2} b \,x^{2 n} {\mathrm e}^{\lambda x}\\ f_2^2 f_0 &=x^{2 n} {\mathrm e}^{\lambda x} a^{2} b \lambda \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} x^{n} a u^{\prime \prime }\left (x \right )-\left (-a^{2} b \,x^{2 n} {\mathrm e}^{\lambda x}+\frac {x^{n} n a}{x}\right ) u^{\prime }\left (x \right )+x^{2 n} {\mathrm e}^{\lambda x} a^{2} b \lambda u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{\frac {a b \,{\mathrm e}^{\lambda x} \left (\lambda \textit {\_Y} \left (x \right )+\textit {\_Y}^{\prime }\left (x \right )\right ) x^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {a b \,{\mathrm e}^{\lambda x} \left (\lambda \textit {\_Y} \left (x \right )+\textit {\_Y}^{\prime }\left (x \right )\right ) x^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {a b \,{\mathrm e}^{\lambda x} \left (\lambda \textit {\_Y} \left (x \right )+\textit {\_Y}^{\prime }\left (x \right )\right ) x^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-n}}{a \operatorname {DESol}\left (\left \{\frac {a b \,{\mathrm e}^{\lambda x} \left (\lambda \textit {\_Y} \left (x \right )+\textit {\_Y}^{\prime }\left (x \right )\right ) x^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {a b \,{\mathrm e}^{\lambda x} \left (\lambda \textit {\_Y} \left (x \right )+\textit {\_Y}^{\prime }\left (x \right )\right ) x^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-n}}{a \operatorname {DESol}\left (\left \{\frac {a b \,{\mathrm e}^{\lambda x} \left (\lambda \textit {\_Y} \left (x \right )+\textit {\_Y}^{\prime }\left (x \right )\right ) x^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {a b \,{\mathrm e}^{\lambda x} \left (\lambda \textit {\_Y} \left (x \right )+\textit {\_Y}^{\prime }\left (x \right )\right ) x^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-n}}{a \operatorname {DESol}\left (\left \{\frac {a b \,{\mathrm e}^{\lambda x} \left (\lambda \textit {\_Y} \left (x \right )+\textit {\_Y}^{\prime }\left (x \right )\right ) x^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {a b \,{\mathrm e}^{\lambda x} \left (\lambda \textit {\_Y} \left (x \right )+\textit {\_Y}^{\prime }\left (x \right )\right ) x^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-n}}{a \operatorname {DESol}\left (\left \{\frac {a b \,{\mathrm e}^{\lambda x} \left (\lambda \textit {\_Y} \left (x \right )+\textit {\_Y}^{\prime }\left (x \right )\right ) x^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-a \,x^{n} y^{2}+a b \,x^{n} {\mathrm e}^{\lambda x} y=b \lambda \,{\mathrm e}^{\lambda x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=a \,x^{n} y^{2}-a b \,x^{n} {\mathrm e}^{\lambda x} y+b \lambda \,{\mathrm e}^{\lambda x} \end {array} \]
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 sub-methods: trying Riccati_symmetries trying Riccati to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = -(exp(lambda*x)*x^n*a*b*x-n)*(diff(y(x), x))/x-b*a*lambda*exp(lambda*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 -> 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 a symmetry of the form [xi=0, eta=F(x)] trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients <- unable to find a useful change of variables trying a symmetry of the form [xi=0, eta=F(x)] trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying 2nd order, integrating factor of the form mu(x,y) 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 -> 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 a symmetry of the form [xi=0, eta=F(x)] trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients <- unable to find a useful change of variables trying a symmetry of the form [xi=0, eta=F(x)] trying to convert to an ODE of Bessel type -> Trying a change of variables to reduce to Bernoulli -> Calling odsolve with the ODE`, diff(y(x), x)-(a*x^n*y(x)^2+y(x)-b*a*exp(lambda*x)*x^n*y(x)*x+exp(lambda*x)*b*lambda*x^2)/x, y( 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 sub-methods: trying Riccati_symmetries trying inverse_Riccati trying 1st order ODE linearizable_by_differentiation -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] trying inverse_Riccati trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 4 `, `-> Computing symmetries using: way = 2 `, `-> Computing symmetries using: way = 6`
✗ Solution by Maple
dsolve(diff(y(x),x)=a*x^n*y(x)^2-a*b*x^n*exp(lambda*x)*y(x)+b*lambda*exp(lambda*x),y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 53.05 (sec). Leaf size: 190
DSolve[y'[x]==a*x^n*y[x]^2-a*b*x^n*Exp[\[Lambda]*x]*y[x]+b*\[Lambda]*Exp[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {b e^{2 \lambda x} \left (\int _1^{e^{x \lambda }}\frac {\exp \left (\frac {a b \Gamma (n+1,-\log (K[1])) (-\log (K[1]))^{-n} \left (\frac {\log (K[1])}{\lambda }\right )^n}{\lambda }\right )}{K[1]^2}dK[1]+c_1\right )}{e^{\lambda x} \int _1^{e^{x \lambda }}\frac {\exp \left (\frac {a b \Gamma (n+1,-\log (K[1])) (-\log (K[1]))^{-n} \left (\frac {\log (K[1])}{\lambda }\right )^n}{\lambda }\right )}{K[1]^2}dK[1]+\exp \left (\frac {a b \left (-\log \left (e^{\lambda x}\right )\right )^{-n} \left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right )^n \Gamma \left (n+1,-\log \left (e^{x \lambda }\right )\right )}{\lambda }\right )+c_1 e^{\lambda x}} \\ y(x)\to b e^{\lambda x} \\ \end{align*}