Internal problem ID [10427]
Internal file name [OUTPUT/9375_Monday_June_06_2022_02_19_44_PM_77437593/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: 20.
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]
\[ \boxed {\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime }-y^{2}-k \,{\mathrm e}^{\nu x} y=-m^{2}+k m \,{\mathrm e}^{\nu x}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {y^{2}+k \,{\mathrm e}^{\nu x} y -m^{2}+k m \,{\mathrm e}^{\nu x}}{{\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \frac {k m \,{\mathrm e}^{\nu x}}{{\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c}+\frac {k \,{\mathrm e}^{\nu x} y}{{\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c}-\frac {m^{2}}{{\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c}+\frac {y^{2}}{{\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {-m^{2}+k m \,{\mathrm e}^{\nu x}}{{\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c}\), \(f_1(x)=\frac {k \,{\mathrm e}^{\nu x}}{{\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c}\) and \(f_2(x)=\frac {1}{{\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {u}{{\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c}} \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 {a \lambda \,{\mathrm e}^{\lambda x}+b \mu \,{\mathrm e}^{\mu x}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}\\ f_1 f_2 &=\frac {k \,{\mathrm e}^{\nu x}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}\\ f_2^2 f_0 &=\frac {-m^{2}+k m \,{\mathrm e}^{\nu x}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \frac {u^{\prime \prime }\left (x \right )}{{\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c}-\left (-\frac {a \lambda \,{\mathrm e}^{\lambda x}+b \mu \,{\mathrm e}^{\mu x}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}+\frac {k \,{\mathrm e}^{\nu x}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}\right ) u^{\prime }\left (x \right )+\frac {\left (-m^{2}+k m \,{\mathrm e}^{\nu x}\right ) u \left (x \right )}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{\frac {\left (2 \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) \left (\lambda +\mu \right )\right ) b a \,{\mathrm e}^{x \left (\lambda +\mu \right )}-k a \textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{x \left (\lambda +\nu \right )}-k \textit {\_Y}^{\prime }\left (x \right ) b \,{\mathrm e}^{x \left (\mu +\nu \right )}+a^{2} \left (\textit {\_Y}^{\prime }\left (x \right ) \lambda +\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \lambda x}+b^{2} \left (\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \mu x}+2 \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+\frac {c}{2}\right ) c \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\lambda x} a c \lambda +\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\mu x} b c \mu +k \left (\textit {\_Y} \left (x \right ) m -c \textit {\_Y}^{\prime }\left (x \right )\right ) {\mathrm e}^{\nu x}-\textit {\_Y} \left (x \right ) m^{2}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\left (2 \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) \left (\lambda +\mu \right )\right ) b a \,{\mathrm e}^{x \left (\lambda +\mu \right )}-k a \textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{x \left (\lambda +\nu \right )}-k \textit {\_Y}^{\prime }\left (x \right ) b \,{\mathrm e}^{x \left (\mu +\nu \right )}+a^{2} \left (\textit {\_Y}^{\prime }\left (x \right ) \lambda +\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \lambda x}+b^{2} \left (\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \mu x}+2 \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+\frac {c}{2}\right ) c \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\lambda x} a c \lambda +\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\mu x} b c \mu +k \left (\textit {\_Y} \left (x \right ) m -c \textit {\_Y}^{\prime }\left (x \right )\right ) {\mathrm e}^{\nu x}-\textit {\_Y} \left (x \right ) m^{2}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\left (2 \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) \left (\lambda +\mu \right )\right ) b a \,{\mathrm e}^{x \left (\lambda +\mu \right )}-k a \textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{x \left (\lambda +\nu \right )}-k \textit {\_Y}^{\prime }\left (x \right ) b \,{\mathrm e}^{x \left (\mu +\nu \right )}+a^{2} \left (\textit {\_Y}^{\prime }\left (x \right ) \lambda +\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \lambda x}+b^{2} \left (\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \mu x}+2 \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+\frac {c}{2}\right ) c \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\lambda x} a c \lambda +\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\mu x} b c \mu +k \left (\textit {\_Y} \left (x \right ) m -c \textit {\_Y}^{\prime }\left (x \right )\right ) {\mathrm e}^{\nu x}-\textit {\_Y} \left (x \right ) m^{2}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )}{\operatorname {DESol}\left (\left \{\frac {\left (2 \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) \left (\lambda +\mu \right )\right ) b a \,{\mathrm e}^{x \left (\lambda +\mu \right )}-k a \textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{x \left (\lambda +\nu \right )}-k \textit {\_Y}^{\prime }\left (x \right ) b \,{\mathrm e}^{x \left (\mu +\nu \right )}+a^{2} \left (\textit {\_Y}^{\prime }\left (x \right ) \lambda +\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \lambda x}+b^{2} \left (\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \mu x}+2 \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+\frac {c}{2}\right ) c \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\lambda x} a c \lambda +\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\mu x} b c \mu +k \left (\textit {\_Y} \left (x \right ) m -c \textit {\_Y}^{\prime }\left (x \right )\right ) {\mathrm e}^{\nu x}-\textit {\_Y} \left (x \right ) m^{2}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}\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 {\left (2 \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) \left (\lambda +\mu \right )\right ) b a \,{\mathrm e}^{x \left (\lambda +\mu \right )}-k a \textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{x \left (\lambda +\nu \right )}-k \textit {\_Y}^{\prime }\left (x \right ) b \,{\mathrm e}^{x \left (\mu +\nu \right )}+a^{2} \left (\textit {\_Y}^{\prime }\left (x \right ) \lambda +\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \lambda x}+b^{2} \left (\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \mu x}+2 \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+\frac {c}{2}\right ) c \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\lambda x} a c \lambda +\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\mu x} b c \mu +k \left (\textit {\_Y} \left (x \right ) m -c \textit {\_Y}^{\prime }\left (x \right )\right ) {\mathrm e}^{\nu x}-\textit {\_Y} \left (x \right ) m^{2}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )}{\operatorname {DESol}\left (\left \{\frac {\left (2 \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) \left (\lambda +\mu \right )\right ) b a \,{\mathrm e}^{x \left (\lambda +\mu \right )}-k a \textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{x \left (\lambda +\nu \right )}-k \textit {\_Y}^{\prime }\left (x \right ) b \,{\mathrm e}^{x \left (\mu +\nu \right )}+a^{2} \left (\textit {\_Y}^{\prime }\left (x \right ) \lambda +\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \lambda x}+b^{2} \left (\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \mu x}+2 \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+\frac {c}{2}\right ) c \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\lambda x} a c \lambda +\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\mu x} b c \mu +k \left (\textit {\_Y} \left (x \right ) m -c \textit {\_Y}^{\prime }\left (x \right )\right ) {\mathrm e}^{\nu x}-\textit {\_Y} \left (x \right ) m^{2}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}\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 {\left (2 \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) \left (\lambda +\mu \right )\right ) b a \,{\mathrm e}^{x \left (\lambda +\mu \right )}-k a \textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{x \left (\lambda +\nu \right )}-k \textit {\_Y}^{\prime }\left (x \right ) b \,{\mathrm e}^{x \left (\mu +\nu \right )}+a^{2} \left (\textit {\_Y}^{\prime }\left (x \right ) \lambda +\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \lambda x}+b^{2} \left (\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \mu x}+2 \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+\frac {c}{2}\right ) c \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\lambda x} a c \lambda +\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\mu x} b c \mu +k \left (\textit {\_Y} \left (x \right ) m -c \textit {\_Y}^{\prime }\left (x \right )\right ) {\mathrm e}^{\nu x}-\textit {\_Y} \left (x \right ) m^{2}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )}{\operatorname {DESol}\left (\left \{\frac {\left (2 \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) \left (\lambda +\mu \right )\right ) b a \,{\mathrm e}^{x \left (\lambda +\mu \right )}-k a \textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{x \left (\lambda +\nu \right )}-k \textit {\_Y}^{\prime }\left (x \right ) b \,{\mathrm e}^{x \left (\mu +\nu \right )}+a^{2} \left (\textit {\_Y}^{\prime }\left (x \right ) \lambda +\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \lambda x}+b^{2} \left (\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \mu x}+2 \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+\frac {c}{2}\right ) c \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\lambda x} a c \lambda +\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\mu x} b c \mu +k \left (\textit {\_Y} \left (x \right ) m -c \textit {\_Y}^{\prime }\left (x \right )\right ) {\mathrm e}^{\nu x}-\textit {\_Y} \left (x \right ) m^{2}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}\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 {\left (2 \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) \left (\lambda +\mu \right )\right ) b a \,{\mathrm e}^{x \left (\lambda +\mu \right )}-k a \textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{x \left (\lambda +\nu \right )}-k \textit {\_Y}^{\prime }\left (x \right ) b \,{\mathrm e}^{x \left (\mu +\nu \right )}+a^{2} \left (\textit {\_Y}^{\prime }\left (x \right ) \lambda +\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \lambda x}+b^{2} \left (\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \mu x}+2 \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+\frac {c}{2}\right ) c \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\lambda x} a c \lambda +\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\mu x} b c \mu +k \left (\textit {\_Y} \left (x \right ) m -c \textit {\_Y}^{\prime }\left (x \right )\right ) {\mathrm e}^{\nu x}-\textit {\_Y} \left (x \right ) m^{2}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )}{\operatorname {DESol}\left (\left \{\frac {\left (2 \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) \left (\lambda +\mu \right )\right ) b a \,{\mathrm e}^{x \left (\lambda +\mu \right )}-k a \textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{x \left (\lambda +\nu \right )}-k \textit {\_Y}^{\prime }\left (x \right ) b \,{\mathrm e}^{x \left (\mu +\nu \right )}+a^{2} \left (\textit {\_Y}^{\prime }\left (x \right ) \lambda +\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \lambda x}+b^{2} \left (\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \mu x}+2 \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+\frac {c}{2}\right ) c \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\lambda x} a c \lambda +\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\mu x} b c \mu +k \left (\textit {\_Y} \left (x \right ) m -c \textit {\_Y}^{\prime }\left (x \right )\right ) {\mathrm e}^{\nu x}-\textit {\_Y} \left (x \right ) m^{2}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime }-y^{2}-k \,{\mathrm e}^{\nu x} y=-m^{2}+k m \,{\mathrm e}^{\nu x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{2}+k \,{\mathrm e}^{\nu x} y-m^{2}+k m \,{\mathrm e}^{\nu x}}{{\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c} \end {array} \]
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: <- Riccati particular case Kamke (b) successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 202
dsolve((a*exp(lambda*x)+b*exp(mu*x)+c)*diff(y(x),x)=y(x)^2+k*exp(nu*x)*y(x)-m^2+k*m*exp(nu*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {-m \left (\int \frac {{\mathrm e}^{k \left (\int \frac {{\mathrm e}^{\nu x}}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )-2 m \left (\int \frac {1}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )}}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )+c_{1} m -{\mathrm e}^{k \left (\int \frac {{\mathrm e}^{\nu x}}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )-2 m \left (\int \frac {1}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )}}{\int \frac {{\mathrm e}^{k \left (\int \frac {{\mathrm e}^{\nu x}}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )-2 m \left (\int \frac {1}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )}}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 16.545 (sec). Leaf size: 358
DSolve[(a*Exp[\[Lambda]*x]+b*Exp[\[Mu]*x]+c)*y'[x]==y[x]^2+k*Exp[\[Nu]*x]*y[x]-m^2+k*m*Exp[\[Nu]*x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x-\frac {\exp \left (-\int _1^{K[2]}-\frac {e^{\nu K[1]} k-2 m}{e^{\lambda K[1]} a+b e^{\mu K[1]}+c}dK[1]\right ) \left (e^{\nu K[2]} k-m+y(x)\right )}{\left (e^{\lambda K[2]} a+b e^{\mu K[2]}+c\right ) k \nu (m+y(x))}dK[2]+\int _1^{y(x)}\left (\frac {\exp \left (-\int _1^x-\frac {e^{\nu K[1]} k-2 m}{e^{\lambda K[1]} a+b e^{\mu K[1]}+c}dK[1]\right )}{k \nu (m+K[3])^2}-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}-\frac {e^{\nu K[1]} k-2 m}{e^{\lambda K[1]} a+b e^{\mu K[1]}+c}dK[1]\right ) \left (e^{\nu K[2]} k-m+K[3]\right )}{\left (e^{\lambda K[2]} a+b e^{\mu K[2]}+c\right ) k \nu (m+K[3])^2}-\frac {\exp \left (-\int _1^{K[2]}-\frac {e^{\nu K[1]} k-2 m}{e^{\lambda K[1]} a+b e^{\mu K[1]}+c}dK[1]\right )}{\left (e^{\lambda K[2]} a+b e^{\mu K[2]}+c\right ) k \nu (m+K[3])}\right )dK[2]\right )dK[3]=c_1,y(x)\right ] \]