2.24.48 Problem 74

Internal problem ID [13612]
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.
Problem number : 74
Date solved : Friday, December 19, 2025 at 08:40:39 AM
CAS classification : [[_Abel, `2nd type`, `class A`]]

\begin{align*} y y^{\prime }+a \left (1+2 \sqrt {x}\, b \right ) {\mathrm e}^{2 \sqrt {x}\, b} y&=-a^{2} b \,x^{{3}/{2}} {\mathrm e}^{4 \sqrt {x}\, b} \\ \end{align*}

Unknown ode type.

2.24.48.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 253

ode:=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)); 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {\sqrt {x}\, \operatorname {BesselI}\left (1, \sqrt {\frac {a}{b^{2} \left ({\mathrm e}^{-2 b \sqrt {x}} y+a x \right )}}\right ) \sqrt {\frac {a}{b^{2} \left ({\mathrm e}^{-2 b \sqrt {x}} y+a x \right )}}\, c_1 b -\operatorname {BesselK}\left (1, -\sqrt {\frac {a}{b^{2} \left ({\mathrm e}^{-2 b \sqrt {x}} y+a x \right )}}\right ) \sqrt {\frac {a}{b^{2} \left ({\mathrm e}^{-2 b \sqrt {x}} y+a x \right )}}\, b \sqrt {x}-\operatorname {BesselI}\left (0, \sqrt {\frac {a}{b^{2} \left ({\mathrm e}^{-2 b \sqrt {x}} y+a x \right )}}\right ) c_1 +\operatorname {BesselK}\left (0, -\sqrt {\frac {a}{b^{2} \left ({\mathrm e}^{-2 b \sqrt {x}} y+a x \right )}}\right )}{\operatorname {BesselI}\left (1, \sqrt {\frac {a}{b^{2} \left ({\mathrm e}^{-2 b \sqrt {x}} y+a x \right )}}\right ) \sqrt {\frac {a}{b^{2} \left ({\mathrm e}^{-2 b \sqrt {x}} y+a x \right )}}\, b \sqrt {x}-\operatorname {BesselI}\left (0, \sqrt {\frac {a}{b^{2} \left ({\mathrm e}^{-2 b \sqrt {x}} y+a x \right )}}\right )} = 0 \]

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
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )+a \left (1+2 b \sqrt {x}\right ) {\mathrm e}^{2 b \sqrt {x}} y \left (x \right )=-a^{2} b \,x^{{3}/{2}} {\mathrm e}^{4 b \sqrt {x}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=-\frac {a \left (1+2 b \sqrt {x}\right ) {\mathrm e}^{2 b \sqrt {x}} y \left (x \right )+a^{2} b \,x^{{3}/{2}} {\mathrm e}^{4 b \sqrt {x}}}{y \left (x \right )} \end {array} \]

2.24.48.2 ✗ Mathematica

ode=y[x]*D[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)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.24.48.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(a**2*b*x**(3/2)*exp(4*b*sqrt(x)) + a*(2*b*sqrt(x) + 1)*y(x)*exp(2*b*sqrt(x)) + y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out