Internal problem ID [10805]
Internal file name [OUTPUT/9753_Wednesday_June_08_2022_05_54_10_PM_38000994/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: 69.
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.
\[ \boxed {y y^{\prime }-\left (a \,{\mathrm e}^{\lambda x}+b \right ) y=c \left (a^{2} {\mathrm e}^{2 \lambda x}+a b \left (\lambda x +1\right ) {\mathrm e}^{\lambda x}+b^{2} \lambda x \right )} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-\left (a \,{\mathrm e}^{\lambda x}+b \right ) y=c \left (a^{2} {\mathrm e}^{2 \lambda x}+a b \left (\lambda x +1\right ) {\mathrm e}^{\lambda x}+b^{2} \lambda x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\left (a \,{\mathrm e}^{\lambda x}+b \right ) y+c \left (a^{2} {\mathrm e}^{2 \lambda x}+a b \left (\lambda x +1\right ) {\mathrm e}^{\lambda x}+b^{2} \lambda x \right )}{y} \end {array} \]
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 <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 257
dsolve(y(x)*diff(y(x),x)=(a*exp(lambda*x)+b)*y(x)+c*(a^2*exp(2*lambda*x)+a*b*(lambda*x+1)*exp(lambda*x)+b^2*lambda*x),y(x), singsol=all)
\[ \frac {\sqrt {\left (4 c \lambda +1\right ) \left (3 c \lambda +1\right )^{2}}\, \left (\frac {c \lambda }{2}+\frac {1}{6}\right ) \ln \left (\frac {\left (3 c \lambda +1\right )^{2} \left (b^{2} c \,\lambda ^{2} x^{2}+2 \,{\mathrm e}^{x \lambda } a b c \lambda x +{\mathrm e}^{2 x \lambda } a^{2} c +b \lambda x y \left (x \right )+a \,{\mathrm e}^{x \lambda } y \left (x \right )-\lambda y \left (x \right )^{2}\right ) c}{\left (9 c \lambda +2\right ) y \left (x \right )^{2}}\right )-3 \left (c \lambda +\frac {1}{3}\right )^{2} \operatorname {arctanh}\left (\frac {\left (3 c \lambda +1\right ) \left (2 b c \lambda x +2 \,{\mathrm e}^{x \lambda } a c +y \left (x \right )\right )}{\sqrt {\left (4 c \lambda +1\right ) \left (3 c \lambda +1\right )^{2}}\, y \left (x \right )}\right )+\sqrt {\left (4 c \lambda +1\right ) \left (3 c \lambda +1\right )^{2}}\, \left (\left (-c \lambda -\frac {1}{3}\right ) \ln \left (\frac {\left (3 c \lambda +1\right ) \left (b \lambda x +{\mathrm e}^{x \lambda } a \right ) c}{y \left (x \right )}\right )+\left (c \lambda +\frac {1}{3}\right ) \ln \left (b \lambda x +{\mathrm e}^{x \lambda } a \right )-c_{1} c \lambda \right )}{\sqrt {\left (4 c \lambda +1\right ) \left (3 c \lambda +1\right )^{2}}\, c \lambda } = 0 \]
✓ Solution by Mathematica
Time used: 0.494 (sec). Leaf size: 134
DSolve[y[x]*y'[x]==(a*Exp[\[Lambda]*x]+b)*y[x]+c*(a^2*Exp[2*\[Lambda]*x]+a*b*(\[Lambda]*x+1)*Exp[\[Lambda]*x]+b^2*\[Lambda]*x),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {\frac {2 \arctan \left (\frac {\frac {2 c \lambda y(x)}{a c e^{\lambda x}+b c \lambda x}-1}{\sqrt {-4 c \lambda -1}}\right )}{\sqrt {-4 c \lambda -1}}+\log \left (-\frac {c \lambda y(x)^2}{\left (a c e^{\lambda x}+b c \lambda x\right )^2}+\frac {y(x)}{a c e^{\lambda x}+b c \lambda x}+1\right )}{2 c \lambda }=\frac {\log \left (a c e^{\lambda x}+b c \lambda x\right )}{c \lambda }+c_1,y(x)\right ] \]