Internal problem ID [10731]
Internal file name [OUTPUT/9679_Monday_June_06_2022_03_21_45_PM_42964674/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 1.3.2. Equations of
the form \(y y'=f(x) y+1\)
Problem number: 7.
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.
\[ \boxed {y y^{\prime }-{\mathrm e}^{\lambda x} y a=1} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-{\mathrm e}^{\lambda x} y a =1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {{\mathrm e}^{\lambda x} y a +1}{y} \end {array} \]
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: 83
dsolve(y(x)*diff(y(x),x)=a*exp(lambda*x)*y(x)+1,y(x), singsol=all)
\[ c_{1} +a \,\operatorname {erf}\left (\frac {\left (\lambda y \left (x \right )-{\mathrm e}^{x \lambda } a \right ) \sqrt {2}}{2 \sqrt {-\lambda }}\right ) \sqrt {2}\, \sqrt {\pi }-2 \sqrt {-\lambda }\, {\mathrm e}^{\frac {y \left (x \right )^{2} \lambda ^{2}-2 y \left (x \right ) {\mathrm e}^{x \lambda } a \lambda +a^{2} {\mathrm e}^{2 x \lambda }-2 x \,\lambda ^{2}}{2 \lambda }} = 0 \]
✓ Solution by Mathematica
Time used: 1.687 (sec). Leaf size: 83
DSolve[y[x]*y'[x]==a*Exp[\[Lambda]*x]*y[x]+1,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {a e^{\lambda x}}{\sqrt {\lambda }}=\frac {2 e^{\frac {\left (a e^{\lambda x}-\lambda y(x)\right )^2}{2 \lambda }}}{\sqrt {2 \pi } \text {erfi}\left (\frac {\lambda y(x)-a e^{\lambda x}}{\sqrt {2} \sqrt {\lambda }}\right )+2 c_1},y(x)\right ] \]