Internal problem ID [10806]
Internal file name [OUTPUT/9754_Wednesday_June_08_2022_05_54_12_PM_60456762/index.tex]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.3-2. Equations
of the form \(y y'=f_1(x) y+f_0(x)\)
Problem number: 70.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_Abel, `2nd type`, `class A`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-{\mathrm e}^{\lambda x} \left (2 a x \lambda +a +b \right ) y=-{\mathrm e}^{2 \lambda x} \left (a^{2} \lambda \,x^{2}+a b x +c \right )} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-{\mathrm e}^{\lambda x} \left (2 a x \lambda +a +b \right ) y=-{\mathrm e}^{2 \lambda x} \left (a^{2} \lambda \,x^{2}+a b x +c \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {{\mathrm e}^{\lambda x} \left (2 a x \lambda +a +b \right ) y-{\mathrm e}^{2 \lambda x} \left (a^{2} \lambda \,x^{2}+a b x +c \right )}{y} \end {array} \]
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 trying Abel Looking for potential symmetries Looking for potential symmetries <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 118
dsolve(y(x)*diff(y(x),x)=exp(lambda*x)*(2*a*lambda*x+a+b)*y(x)-exp(2*lambda*x)*(a^2*lambda*x^2+a*b*x+c),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\tan \left (\frac {\operatorname {RootOf}\left (2 a x \lambda \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}-\sqrt {-\frac {b^{2}-4 c \lambda }{a^{2}}}\, \tan \left (\frac {\textit {\_a} \sqrt {-\frac {b^{2}-4 c \lambda }{a^{2}}}}{2}\right ) \textit {\_Z} a +b \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+2 c_{1} a \,{\mathrm e}^{\textit {\_a}}\right ) \sqrt {\frac {-b^{2}+4 c \lambda }{a^{2}}}}{2}\right ) a \sqrt {\frac {-b^{2}+4 c \lambda }{a^{2}}}+2 x a \lambda +b \right ) {\mathrm e}^{x \lambda }}{2 \lambda } \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]==Exp[\[Lambda]*x]*(2*a*\[Lambda]*x+a+b)*y[x]-Exp[2*\[Lambda]*x]*(a^2*\[Lambda]*x^2+a*b*x+c),y[x],x,IncludeSingularSolutions -> True]
Not solved