23.4.30 problem 30

Internal problem ID [6332]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 30
Date solved : Friday, October 03, 2025 at 02:01:18 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime }&=a +4 b^{2} y+3 b y^{2}+3 y y^{\prime } \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 343

ode:=diff(diff(y(x),x),x) = a+4*b^2*y(x)+3*b*y(x)^2+3*y(x)*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

\[ -2 b^{2} \int _{}^{y}\frac {1}{-4 \textit {\_a} \,b^{3}-3 \textit {\_a}^{2} b^{2}-a b +\operatorname {RootOf}\left (\left (-\operatorname {BesselK}\left (\frac {-4 b^{3}+3 a}{2 b \sqrt {4 b^{4}-3 a b}}, \frac {\sqrt {3}\, \sqrt {\textit {\_Z}}}{2 b^{2}}\right ) c_1 -\operatorname {BesselI}\left (-\frac {-4 b^{3}+3 a}{2 b \sqrt {4 b^{4}-3 a b}}, \frac {\sqrt {3}\, \sqrt {\textit {\_Z}}}{2 b^{2}}\right )\right ) \sqrt {4 b^{4}-3 a b}+\left (\operatorname {BesselK}\left (\frac {4 b^{3}+2 b \sqrt {4 b^{4}-3 a b}-3 a}{2 b \sqrt {4 b^{4}-3 a b}}, \frac {\sqrt {3}\, \sqrt {\textit {\_Z}}}{2 b^{2}}\right ) c_1 -\operatorname {BesselI}\left (\frac {4 b^{3}+2 b \sqrt {4 b^{4}-3 a b}-3 a}{2 b \sqrt {4 b^{4}-3 a b}}, \frac {\sqrt {3}\, \sqrt {\textit {\_Z}}}{2 b^{2}}\right )\right ) \sqrt {3}\, \sqrt {\textit {\_Z}}+2 \left (b +\frac {3 \textit {\_a}}{2}\right ) \left (\operatorname {BesselK}\left (\frac {-4 b^{3}+3 a}{2 b \sqrt {4 b^{4}-3 a b}}, \frac {\sqrt {3}\, \sqrt {\textit {\_Z}}}{2 b^{2}}\right ) c_1 +\operatorname {BesselI}\left (-\frac {-4 b^{3}+3 a}{2 b \sqrt {4 b^{4}-3 a b}}, \frac {\sqrt {3}\, \sqrt {\textit {\_Z}}}{2 b^{2}}\right )\right ) b \right )}d \textit {\_a} -x -c_2 = 0 \]

✓ Mathematica. Time used: 122.928 (sec). Leaf size: 2742

ode=D[y[x],{x,2}] == a + 4*b^2*y[x] + 3*b*y[x]^2 + 3*y[x]*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Too large to display

✗ Sympy

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

NotImplementedError : The given ODE -(-a - b*(4*b + 3*y(x))*y(x) + Derivative(y(x), (x, 2)))/(3*y(x)