Internal problem ID [10810]
Internal file name [OUTPUT/9758_Thursday_June_09_2022_12_57_33_AM_22585268/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: 74.
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.
\[ \boxed {y y^{\prime }+a \left (1+2 b \sqrt {x}\right ) {\mathrm e}^{2 b \sqrt {x}} y=-a^{2} b \,x^{\frac {3}{2}} {\mathrm e}^{4 b \sqrt {x}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }+a \left (1+2 b \sqrt {x}\right ) {\mathrm e}^{2 b \sqrt {x}} y=-a^{2} b \,x^{\frac {3}{2}} {\mathrm e}^{4 b \sqrt {x}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {a \left (1+2 b \sqrt {x}\right ) {\mathrm e}^{2 b \sqrt {x}} y+a^{2} b \,x^{\frac {3}{2}} {\mathrm e}^{4 b \sqrt {x}}}{y} \end {array} \]
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 Looking for potential symmetries Looking for potential symmetries <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 307
dsolve(y(x)*diff(y(x),x)+a*(1+2*b*x^(1/2))*exp(2*b*x^(1/2))*y(x)=-a^2*b*x^(3/2)*exp(4*b*x^(1/2)),y(x), singsol=all)
\[ \frac {\sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\, \sqrt {x}\, \operatorname {BesselI}\left (1, \sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\right ) c_{1} b -\operatorname {BesselK}\left (1, -\sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\, b \sqrt {x}-\operatorname {BesselI}\left (0, \sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\right ) c_{1} +\operatorname {BesselK}\left (0, -\sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\right )}{\operatorname {BesselI}\left (1, \sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\, b \sqrt {x}-\operatorname {BesselI}\left (0, \sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]+a*(1+2*b*x^(1/2))*Exp[2*b*x^(1/2)]*y[x]==-a^2*b*x^(3/2)*exp(4*b*x^(1/2)),y[x],x,IncludeSingularSolutions -> True]
Not solved