23.4.158 problem 158
Internal
problem
ID
[6460]
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
:
158
Date
solved
:
Friday, October 03, 2025 at 02:05:44 AM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]
\begin{align*} y y^{\prime \prime }&=\operatorname {a2} y^{2}+\operatorname {a3} y^{1+a}+\operatorname {a1} y y^{\prime }+a {y^{\prime }}^{2} \end{align*}
✓ Maple. Time used: 0.075 (sec). Leaf size: 126
ode:=y(x)*diff(diff(y(x),x),x) = a2*y(x)^2+a3*y(x)^(a+1)+a1*y(x)*diff(y(x),x)+a*diff(y(x),x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = {\mathrm e}^{-\frac {\left (\operatorname {a1} -\sqrt {\left (-4 a +4\right ) \operatorname {a2} +\operatorname {a1}^{2}}\right ) x}{2 a -2}} \left (\frac {\left (-4 a +4\right ) \operatorname {a2}^{3}+\operatorname {a1}^{2} \operatorname {a2}^{2}}{\left (-\operatorname {a3} \,{\mathrm e}^{\frac {\left (-\operatorname {a1} +\sqrt {\left (-4 a +4\right ) \operatorname {a2} +\operatorname {a1}^{2}}\right ) x}{2}} \sqrt {\left (-4 a +4\right ) \operatorname {a2} +\operatorname {a1}^{2}}+\operatorname {a2} \left ({\mathrm e}^{x \sqrt {\left (-4 a +4\right ) \operatorname {a2} +\operatorname {a1}^{2}}} c_1 -c_2 \right ) \left (a -1\right )\right )^{2}}\right )^{\frac {1}{2 a -2}}
\]
✓ Mathematica. Time used: 60.802 (sec). Leaf size: 534
ode=y[x]*D[y[x],{x,2}] == a2*y[x]^2 + a3*y[x]^(1 + a) + a1*y[x]*D[y[x],x] + a*D[y[x],x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to 2^{\frac {1}{1-a}} \left (-\frac {e^{-\frac {2 (a-1) \text {a2} x}{\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}}} \left (\text {a1}^2 \left (\text {a3} \left (-\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}\right ) e^{\frac {2 (a-1) \text {a2} x}{\sqrt {\text {a1}^2-4 (a-1) \text {a2}}-\text {a1}}}-(a-1) \text {a2} c_1 \exp \left (\frac {x \left (\text {a1} \sqrt {\text {a1}^2-4 (a-1) \text {a2}}+4 (a-1) \text {a2}-\text {a1}^2\right )}{\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}}\right )+\text {a2} c_2 \sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}\right )+(a-1) \text {a1} \text {a2} \left (-4 \text {a3} e^{\frac {2 (a-1) \text {a2} x}{\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}}}+c_1 \sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}} \exp \left (\frac {x \left (\text {a1} \sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}+4 (a-1) \text {a2}-\text {a1}^2\right )}{\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}}\right )+4 \text {a2} c_2\right )+2 (a-1) \text {a2} \left (\text {a3} \sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}} e^{\frac {2 (a-1) \text {a2} x}{\sqrt {\text {a1}^2-4 (a-1) \text {a2}}-\text {a1}}}+(a-1) \text {a2} c_1 \exp \left (\frac {x \left (\text {a1} \sqrt {\text {a1}^2-4 (a-1) \text {a2}}+4 (a-1) \text {a2}-\text {a1}^2\right )}{\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}}\right )-\text {a2} c_2 \sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}\right )+\text {a1}^3 \left (\text {a3} e^{\frac {2 (a-1) \text {a2} x}{\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}}}-\text {a2} c_2\right )\right )}{\text {a2} \left (\text {a1} \sqrt {\text {a1}^2-4 (a-1) \text {a2}}+4 (a-1) \text {a2}-\text {a1}^2\right ) \left (\sqrt {-4 a \text {a2}+\text {a1}^2+4 \text {a2}}-\text {a1}\right )}\right ){}^{\frac {1}{1-a}} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
a1 = symbols("a1")
a2 = symbols("a2")
a3 = symbols("a3")
y = Function("y")
ode = Eq(-a*Derivative(y(x), x)**2 - a1*y(x)*Derivative(y(x), x) - a2*y(x)**2 - a3*y(x)**(a + 1) + y(x)*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (-a1*y(x) + sqrt(-4*a*a2*y(x)**2 - 4*a*a3*