Internal problem ID [10803]
Internal file name [OUTPUT/9751_Wednesday_June_08_2022_05_54_05_PM_43706231/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: 67.
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.
\[ \boxed {y y^{\prime }-\left (a \,{\mathrm e}^{x}+b \right ) y=c \,{\mathrm e}^{2 x}-a b \,{\mathrm e}^{x}-b^{2}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-\left (a \,{\mathrm e}^{x}+b \right ) y=c \,{\mathrm e}^{2 x}-a b \,{\mathrm e}^{x}-b^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\left (a \,{\mathrm e}^{x}+b \right ) y+c \,{\mathrm e}^{2 x}-a b \,{\mathrm e}^{x}-b^{2}}{y} \end {array} \]
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 found: 2 potential symmetries. Proceeding with integration step <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 153
dsolve(y(x)*diff(y(x),x)=(a*exp(x)+b)*y(x)+c*exp(2*x)-a*b*exp(x)-b^2,y(x), singsol=all)
\[ \sqrt {\frac {c \,{\mathrm e}^{2 x}-\left (b -y \left (x \right )\right ) \left (a \,{\mathrm e}^{x}+b -y \left (x \right )\right )}{\left (b -y \left (x \right )\right )^{2}}}\, y \left (x \right ) {\mathrm e}^{-\frac {a \,\operatorname {arctanh}\left (\frac {\left (b -y \left (x \right )\right ) a -2 \,{\mathrm e}^{x} c}{\sqrt {a^{2}+4 c}\, \left (b -y \left (x \right )\right )}\right )}{\sqrt {a^{2}+4 c}}}-b \left (\int _{}^{\frac {{\mathrm e}^{x}}{-b +y \left (x \right )}}\frac {\sqrt {\textit {\_a}^{2} c +a \textit {\_a} -1}\, {\mathrm e}^{-\frac {a \,\operatorname {arctanh}\left (\frac {2 c \textit {\_a} +a}{\sqrt {a^{2}+4 c}}\right )}{\sqrt {a^{2}+4 c}}}}{\textit {\_a}}d \textit {\_a} \right )+c_{1} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]==(a*Exp[x]+b)*y[x]+c*Exp[2*x]-a*b*Exp[x]-b^2,y[x],x,IncludeSingularSolutions -> True]
Not solved