Internal problem ID [10808]
Internal file name [OUTPUT/9756_Thursday_June_09_2022_12_57_29_AM_9550685/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: 72.
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.
\[ \boxed {y y^{\prime }+a \left (2 b x +1\right ) {\mathrm e}^{b x} y=-a^{2} b \,x^{2} {\mathrm e}^{2 b x}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }+a \left (2 b x +1\right ) {\mathrm e}^{b x} y=-a^{2} b \,x^{2} {\mathrm e}^{2 b x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {a \left (2 b x +1\right ) {\mathrm e}^{b x} y+a^{2} b \,x^{2} {\mathrm e}^{2 b x}}{y} \end {array} \]
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: 53
dsolve(y(x)*diff(y(x),x)+a*(1+2*b*x)*exp(b*x)*y(x)=-a^2*b*x^2*exp(2*b*x),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {{\mathrm e}^{b x} a \left (b x \operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}} b x -\operatorname {expIntegral}_{1}\left (-\textit {\_Z} \right )+c_{1} \right )-1\right )}{\operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}} b x -\operatorname {expIntegral}_{1}\left (-\textit {\_Z} \right )+c_{1} \right ) b} \]
✓ Solution by Mathematica
Time used: 0.736 (sec). Leaf size: 59
DSolve[y[x]*y'[x]+a*(1+2*b*x)*Exp[b*x]*y[x]==-a^2*b*x^2*Exp[2*b*x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [b x e^{\frac {a e^{b x}}{a b x e^{b x}+b y(x)}}=\operatorname {ExpIntegralEi}\left (\frac {a e^{b x}}{a b e^{b x} x+b y(x)}\right )+c_1,y(x)\right ] \]