Internal problem ID [10642]
Internal file name [OUTPUT/9590_Monday_June_06_2022_03_11_12_PM_76019185/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 1.2.9. Some Transformations
Problem number: 8.
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]
\[ \boxed {y^{\prime }-y^{2}=-\frac {\lambda ^{2}}{4}+\frac {{\mathrm e}^{2 \lambda x} f \left (\frac {a \,{\mathrm e}^{\lambda x}+b}{c \,{\mathrm e}^{\lambda x}+d}\right )}{\left (c \,{\mathrm e}^{\lambda x}+d \right )^{4}}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {-{\mathrm e}^{4 \lambda x} c^{4} \lambda ^{2}+4 \,{\mathrm e}^{4 \lambda x} c^{4} y^{2}-4 \,{\mathrm e}^{3 \lambda x} c^{3} d \,\lambda ^{2}+16 \,{\mathrm e}^{3 \lambda x} c^{3} d \,y^{2}-6 \,{\mathrm e}^{2 \lambda x} c^{2} d^{2} \lambda ^{2}+24 \,{\mathrm e}^{2 \lambda x} c^{2} d^{2} y^{2}-4 \,{\mathrm e}^{\lambda x} c \,d^{3} \lambda ^{2}+16 \,{\mathrm e}^{\lambda x} c \,d^{3} y^{2}-d^{4} \lambda ^{2}+4 d^{4} y^{2}+4 \,{\mathrm e}^{2 \lambda x} f \left (\frac {a \,{\mathrm e}^{\lambda x}+b}{c \,{\mathrm e}^{\lambda x}+d}\right )}{4 \,{\mathrm e}^{4 \lambda x} c^{4}+16 \,{\mathrm e}^{3 \lambda x} c^{3} d +24 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+16 c \,{\mathrm e}^{\lambda x} d^{3}+4 d^{4}} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -\frac {{\mathrm e}^{4 \lambda x} c^{4} \lambda ^{2}}{4 \left ({\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}\right )}+\frac {{\mathrm e}^{4 \lambda x} c^{4} y^{2}}{{\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}}-\frac {{\mathrm e}^{3 \lambda x} c^{3} d \,\lambda ^{2}}{{\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}}+\frac {4 \,{\mathrm e}^{3 \lambda x} c^{3} d \,y^{2}}{{\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}}-\frac {3 \,{\mathrm e}^{2 \lambda x} c^{2} d^{2} \lambda ^{2}}{2 \left ({\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}\right )}+\frac {6 \,{\mathrm e}^{2 \lambda x} c^{2} d^{2} y^{2}}{{\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}}-\frac {{\mathrm e}^{\lambda x} c \,d^{3} \lambda ^{2}}{{\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}}+\frac {4 \,{\mathrm e}^{\lambda x} c \,d^{3} y^{2}}{{\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}}-\frac {d^{4} \lambda ^{2}}{4 \left ({\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}\right )}+\frac {d^{4} y^{2}}{{\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}}+\frac {{\mathrm e}^{2 \lambda x} f \left (\frac {a \,{\mathrm e}^{\lambda x}+b}{c \,{\mathrm e}^{\lambda x}+d}\right )}{{\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {-{\mathrm e}^{4 \lambda x} c^{4} \lambda ^{2}-4 \,{\mathrm e}^{3 \lambda x} c^{3} d \,\lambda ^{2}-6 \,{\mathrm e}^{2 \lambda x} c^{2} d^{2} \lambda ^{2}-4 \,{\mathrm e}^{\lambda x} c \,d^{3} \lambda ^{2}-d^{4} \lambda ^{2}+4 \,{\mathrm e}^{2 \lambda x} f \left (\frac {a \,{\mathrm e}^{\lambda x}+b}{c \,{\mathrm e}^{\lambda x}+d}\right )}{4 \,{\mathrm e}^{4 \lambda x} c^{4}+16 \,{\mathrm e}^{3 \lambda x} c^{3} d +24 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+16 c \,{\mathrm e}^{\lambda x} d^{3}+4 d^{4}}\), \(f_1(x)=0\) 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' &=\frac {16 \,{\mathrm e}^{4 \lambda x} c^{4} \lambda +48 \,{\mathrm e}^{3 \lambda x} c^{3} d \lambda +48 \,{\mathrm e}^{2 \lambda x} c^{2} d^{2} \lambda +16 \,{\mathrm e}^{\lambda x} c \,d^{3} \lambda }{4 \,{\mathrm e}^{4 \lambda x} c^{4}+16 \,{\mathrm e}^{3 \lambda x} c^{3} d +24 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+16 c \,{\mathrm e}^{\lambda x} d^{3}+4 d^{4}}-\frac {\left (4 \,{\mathrm e}^{4 \lambda x} c^{4}+16 \,{\mathrm e}^{3 \lambda x} c^{3} d +24 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+16 c \,{\mathrm e}^{\lambda x} d^{3}+4 d^{4}\right ) \left (4 \,{\mathrm e}^{4 \lambda x} c^{4} \lambda +12 \,{\mathrm e}^{3 \lambda x} c^{3} d \lambda +12 \,{\mathrm e}^{2 \lambda x} c^{2} d^{2} \lambda +4 \,{\mathrm e}^{\lambda x} c \,d^{3} \lambda \right )}{4 \left ({\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}\right )^{2}}\\ f_1 f_2 &=0\\ f_2^2 f_0 &=\frac {\left (4 \,{\mathrm e}^{4 \lambda x} c^{4}+16 \,{\mathrm e}^{3 \lambda x} c^{3} d +24 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+16 c \,{\mathrm e}^{\lambda x} d^{3}+4 d^{4}\right )^{2} \left (-{\mathrm e}^{4 \lambda x} c^{4} \lambda ^{2}-4 \,{\mathrm e}^{3 \lambda x} c^{3} d \,\lambda ^{2}-6 \,{\mathrm e}^{2 \lambda x} c^{2} d^{2} \lambda ^{2}-4 \,{\mathrm e}^{\lambda x} c \,d^{3} \lambda ^{2}-d^{4} \lambda ^{2}+4 \,{\mathrm e}^{2 \lambda x} f \left (\frac {a \,{\mathrm e}^{\lambda x}+b}{c \,{\mathrm e}^{\lambda x}+d}\right )\right )}{64 \left ({\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}\right )^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} u^{\prime \prime }\left (x \right )-\left (\frac {16 \,{\mathrm e}^{4 \lambda x} c^{4} \lambda +48 \,{\mathrm e}^{3 \lambda x} c^{3} d \lambda +48 \,{\mathrm e}^{2 \lambda x} c^{2} d^{2} \lambda +16 \,{\mathrm e}^{\lambda x} c \,d^{3} \lambda }{4 \,{\mathrm e}^{4 \lambda x} c^{4}+16 \,{\mathrm e}^{3 \lambda x} c^{3} d +24 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+16 c \,{\mathrm e}^{\lambda x} d^{3}+4 d^{4}}-\frac {\left (4 \,{\mathrm e}^{4 \lambda x} c^{4}+16 \,{\mathrm e}^{3 \lambda x} c^{3} d +24 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+16 c \,{\mathrm e}^{\lambda x} d^{3}+4 d^{4}\right ) \left (4 \,{\mathrm e}^{4 \lambda x} c^{4} \lambda +12 \,{\mathrm e}^{3 \lambda x} c^{3} d \lambda +12 \,{\mathrm e}^{2 \lambda x} c^{2} d^{2} \lambda +4 \,{\mathrm e}^{\lambda x} c \,d^{3} \lambda \right )}{4 \left ({\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}\right )^{2}}\right ) u^{\prime }\left (x \right )+\frac {\left (4 \,{\mathrm e}^{4 \lambda x} c^{4}+16 \,{\mathrm e}^{3 \lambda x} c^{3} d +24 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+16 c \,{\mathrm e}^{\lambda x} d^{3}+4 d^{4}\right )^{2} \left (-{\mathrm e}^{4 \lambda x} c^{4} \lambda ^{2}-4 \,{\mathrm e}^{3 \lambda x} c^{3} d \,\lambda ^{2}-6 \,{\mathrm e}^{2 \lambda x} c^{2} d^{2} \lambda ^{2}-4 \,{\mathrm e}^{\lambda x} c \,d^{3} \lambda ^{2}-d^{4} \lambda ^{2}+4 \,{\mathrm e}^{2 \lambda x} f \left (\frac {a \,{\mathrm e}^{\lambda x}+b}{c \,{\mathrm e}^{\lambda x}+d}\right )\right ) u \left (x \right )}{64 \left ({\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}\right )^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{\frac {\textit {\_Y} \left (x \right ) {\mathrm e}^{2 \lambda x} f \left (\frac {a \,{\mathrm e}^{\lambda x}+b}{c \,{\mathrm e}^{\lambda x}+d}\right )+6 \left (-\frac {\textit {\_Y} \left (x \right ) \lambda ^{2}}{4}+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) \left (c^{2} d^{2} {\mathrm e}^{2 \lambda x}+\frac {2 \,{\mathrm e}^{3 \lambda x} c^{3} d}{3}+\frac {{\mathrm e}^{4 \lambda x} c^{4}}{6}+\frac {2 d^{3} \left (c \,{\mathrm e}^{\lambda x}+\frac {d}{4}\right )}{3}\right )}{{\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y} \left (x \right ) {\mathrm e}^{2 \lambda x} f \left (\frac {a \,{\mathrm e}^{\lambda x}+b}{c \,{\mathrm e}^{\lambda x}+d}\right )+6 \left (-\frac {\textit {\_Y} \left (x \right ) \lambda ^{2}}{4}+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) \left (c^{2} d^{2} {\mathrm e}^{2 \lambda x}+\frac {2 \,{\mathrm e}^{3 \lambda x} c^{3} d}{3}+\frac {{\mathrm e}^{4 \lambda x} c^{4}}{6}+\frac {2 d^{3} \left (c \,{\mathrm e}^{\lambda x}+\frac {d}{4}\right )}{3}\right )}{{\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {4 \left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y} \left (x \right ) {\mathrm e}^{2 \lambda x} f \left (\frac {a \,{\mathrm e}^{\lambda x}+b}{c \,{\mathrm e}^{\lambda x}+d}\right )+6 \left (-\frac {\textit {\_Y} \left (x \right ) \lambda ^{2}}{4}+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) \left (c^{2} d^{2} {\mathrm e}^{2 \lambda x}+\frac {2 \,{\mathrm e}^{3 \lambda x} c^{3} d}{3}+\frac {{\mathrm e}^{4 \lambda x} c^{4}}{6}+\frac {2 d^{3} \left (c \,{\mathrm e}^{\lambda x}+\frac {d}{4}\right )}{3}\right )}{{\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \left ({\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}\right )}{\left (4 \,{\mathrm e}^{4 \lambda x} c^{4}+16 \,{\mathrm e}^{3 \lambda x} c^{3} d +24 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+16 c \,{\mathrm e}^{\lambda x} d^{3}+4 d^{4}\right ) \operatorname {DESol}\left (\left \{\frac {\textit {\_Y} \left (x \right ) {\mathrm e}^{2 \lambda x} f \left (\frac {a \,{\mathrm e}^{\lambda x}+b}{c \,{\mathrm e}^{\lambda x}+d}\right )+6 \left (-\frac {\textit {\_Y} \left (x \right ) \lambda ^{2}}{4}+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) \left (c^{2} d^{2} {\mathrm e}^{2 \lambda x}+\frac {2 \,{\mathrm e}^{3 \lambda x} c^{3} d}{3}+\frac {{\mathrm e}^{4 \lambda x} c^{4}}{6}+\frac {2 d^{3} \left (c \,{\mathrm e}^{\lambda x}+\frac {d}{4}\right )}{3}\right )}{{\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y} \left (x \right ) {\mathrm e}^{2 \lambda x} f \left (\frac {a \,{\mathrm e}^{\lambda x}+b}{c \,{\mathrm e}^{\lambda x}+d}\right )+6 \left (-\frac {\textit {\_Y} \left (x \right ) \lambda ^{2}}{4}+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) \left (c^{2} d^{2} {\mathrm e}^{2 \lambda x}+\frac {2 \,{\mathrm e}^{3 \lambda x} c^{3} d}{3}+\frac {{\mathrm e}^{4 \lambda x} c^{4}}{6}+\frac {2 d^{3} \left (c \,{\mathrm e}^{\lambda x}+\frac {d}{4}\right )}{3}\right )}{{\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{\operatorname {DESol}\left (\left \{\frac {\textit {\_Y} \left (x \right ) {\mathrm e}^{2 \lambda x} f \left (\frac {a \,{\mathrm e}^{\lambda x}+b}{c \,{\mathrm e}^{\lambda x}+d}\right )+6 \left (-\frac {\textit {\_Y} \left (x \right ) \lambda ^{2}}{4}+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) \left (c^{2} d^{2} {\mathrm e}^{2 \lambda x}+\frac {2 \,{\mathrm e}^{3 \lambda x} c^{3} d}{3}+\frac {{\mathrm e}^{4 \lambda x} c^{4}}{6}+\frac {2 d^{3} \left (c \,{\mathrm e}^{\lambda x}+\frac {d}{4}\right )}{3}\right )}{{\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y} \left (x \right ) {\mathrm e}^{2 \lambda x} f \left (\frac {a \,{\mathrm e}^{\lambda x}+b}{c \,{\mathrm e}^{\lambda x}+d}\right )+6 \left (-\frac {\textit {\_Y} \left (x \right ) \lambda ^{2}}{4}+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) \left (c^{2} d^{2} {\mathrm e}^{2 \lambda x}+\frac {2 \,{\mathrm e}^{3 \lambda x} c^{3} d}{3}+\frac {{\mathrm e}^{4 \lambda x} c^{4}}{6}+\frac {2 d^{3} \left (c \,{\mathrm e}^{\lambda x}+\frac {d}{4}\right )}{3}\right )}{{\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{\operatorname {DESol}\left (\left \{\frac {\textit {\_Y} \left (x \right ) {\mathrm e}^{2 \lambda x} f \left (\frac {a \,{\mathrm e}^{\lambda x}+b}{c \,{\mathrm e}^{\lambda x}+d}\right )+6 \left (-\frac {\textit {\_Y} \left (x \right ) \lambda ^{2}}{4}+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) \left (c^{2} d^{2} {\mathrm e}^{2 \lambda x}+\frac {2 \,{\mathrm e}^{3 \lambda x} c^{3} d}{3}+\frac {{\mathrm e}^{4 \lambda x} c^{4}}{6}+\frac {2 d^{3} \left (c \,{\mathrm e}^{\lambda x}+\frac {d}{4}\right )}{3}\right )}{{\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y} \left (x \right ) {\mathrm e}^{2 \lambda x} f \left (\frac {a \,{\mathrm e}^{\lambda x}+b}{c \,{\mathrm e}^{\lambda x}+d}\right )+6 \left (-\frac {\textit {\_Y} \left (x \right ) \lambda ^{2}}{4}+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) \left (c^{2} d^{2} {\mathrm e}^{2 \lambda x}+\frac {2 \,{\mathrm e}^{3 \lambda x} c^{3} d}{3}+\frac {{\mathrm e}^{4 \lambda x} c^{4}}{6}+\frac {2 d^{3} \left (c \,{\mathrm e}^{\lambda x}+\frac {d}{4}\right )}{3}\right )}{{\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{\operatorname {DESol}\left (\left \{\frac {\textit {\_Y} \left (x \right ) {\mathrm e}^{2 \lambda x} f \left (\frac {a \,{\mathrm e}^{\lambda x}+b}{c \,{\mathrm e}^{\lambda x}+d}\right )+6 \left (-\frac {\textit {\_Y} \left (x \right ) \lambda ^{2}}{4}+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) \left (c^{2} d^{2} {\mathrm e}^{2 \lambda x}+\frac {2 \,{\mathrm e}^{3 \lambda x} c^{3} d}{3}+\frac {{\mathrm e}^{4 \lambda x} c^{4}}{6}+\frac {2 d^{3} \left (c \,{\mathrm e}^{\lambda x}+\frac {d}{4}\right )}{3}\right )}{{\mathrm e}^{4 \lambda x} c^{4}+4 \,{\mathrm e}^{3 \lambda x} c^{3} d +6 c^{2} d^{2} {\mathrm e}^{2 \lambda x}+4 c \,{\mathrm e}^{\lambda x} d^{3}+d^{4}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 y^{2} \left ({\mathrm e}^{\lambda x}\right )^{4} c^{4}-\left ({\mathrm e}^{\lambda x}\right )^{4} c^{4} \lambda ^{2}-4 y^{\prime } \left ({\mathrm e}^{\lambda x}\right )^{4} c^{4}+16 y^{2} \left ({\mathrm e}^{\lambda x}\right )^{3} c^{3} d -4 \left ({\mathrm e}^{\lambda x}\right )^{3} c^{3} d \,\lambda ^{2}-16 y^{\prime } \left ({\mathrm e}^{\lambda x}\right )^{3} c^{3} d +24 y^{2} \left ({\mathrm e}^{\lambda x}\right )^{2} c^{2} d^{2}-6 \left ({\mathrm e}^{\lambda x}\right )^{2} c^{2} d^{2} \lambda ^{2}-24 y^{\prime } \left ({\mathrm e}^{\lambda x}\right )^{2} c^{2} d^{2}+16 y^{2} {\mathrm e}^{\lambda x} c \,d^{3}-4 \,{\mathrm e}^{\lambda x} c \,d^{3} \lambda ^{2}-16 y^{\prime } {\mathrm e}^{\lambda x} c \,d^{3}+4 y^{2} d^{4}-d^{4} \lambda ^{2}-4 y^{\prime } d^{4}+4 \,{\mathrm e}^{2 \lambda x} f \left (\frac {a \,{\mathrm e}^{\lambda x}+b}{c \,{\mathrm e}^{\lambda x}+d}\right )=0 \\ \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 {-4 y^{2} \left ({\mathrm e}^{\lambda x}\right )^{4} c^{4}+\left ({\mathrm e}^{\lambda x}\right )^{4} c^{4} \lambda ^{2}-16 y^{2} \left ({\mathrm e}^{\lambda x}\right )^{3} c^{3} d +4 \left ({\mathrm e}^{\lambda x}\right )^{3} c^{3} d \,\lambda ^{2}-24 y^{2} \left ({\mathrm e}^{\lambda x}\right )^{2} c^{2} d^{2}+6 \left ({\mathrm e}^{\lambda x}\right )^{2} c^{2} d^{2} \lambda ^{2}-16 y^{2} {\mathrm e}^{\lambda x} c \,d^{3}+4 \,{\mathrm e}^{\lambda x} c \,d^{3} \lambda ^{2}-4 y^{2} d^{4}+d^{4} \lambda ^{2}-4 \,{\mathrm e}^{2 \lambda x} f \left (\frac {a \,{\mathrm e}^{\lambda x}+b}{c \,{\mathrm e}^{\lambda x}+d}\right )}{-4 c^{4} \left ({\mathrm e}^{\lambda x}\right )^{4}-16 c^{3} \left ({\mathrm e}^{\lambda x}\right )^{3} d -24 c^{2} \left ({\mathrm e}^{\lambda x}\right )^{2} d^{2}-16 c \,{\mathrm e}^{\lambda x} d^{3}-4 d^{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 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 trying symmetry patterns for 1st order ODEs -> 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 symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] `, `-> Computing symmetries using: way = HINT -> Calling odsolve with the ODE`, diff(y(x), x)-2*lambda*K[1], y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)+2*lambda*K[1], y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)+4*y(x)*lambda*(2*f((exp(lambda*x)*a+b)/(exp(lambda*x)*c+d))*exp(2*lambda*x)*d^2-2 Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful `, `-> Computing symmetries using: way = HINT -> Calling odsolve with the ODE`, diff(y(x), x)-2*lambda*d^4, y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)+4*(-35*lambda^3*d^4+140*lambda*d^4*x^2+2*y(x)*x)/((lambda-2*x)*(2*x+lambda)), y(x Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)+(-lambda^3*d^4+4*lambda*d^4*x^2+8*y(x)*x)/((lambda-2*x)*(2*x+lambda)), y(x)` Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)+(-21*lambda^3*d^4+84*lambda*d^4*x^2+8*y(x)*x)/((lambda-2*x)*(2*x+lambda)), y(x)` Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)+(2/3)*(-7*lambda^3*d^4+28*lambda*d^4*x^2+12*y(x)*x)/((lambda-2*x)*(2*x+lambda)), Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> trying a symmetry pattern of the form [F(x),G(x)] -> trying a symmetry pattern of the form [F(y),G(y)] -> 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 a symmetry pattern of conformal type`
✗ Solution by Maple
dsolve(diff(y(x),x)=y(x)^2-lambda^2/4+exp(2*lambda*x)/(c*exp(lambda*x)+d)^4*f((a*exp(lambda*x)+b)/(c*exp(lambda*x)+d)),y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==y[x]^2-\[Lambda]^2/4+Exp[2*\[Lambda]*x]/(c*Exp[\[Lambda]*x]+d)^4*f[(a*Exp[\[Lambda]*x]+b)/(c*Exp[\[Lambda]*x]+d)],y[x],x,IncludeSingularSolutions -> True]
Not solved