23.4.243 problem 243
Internal
problem
ID
[6545]
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
:
243
Date
solved
:
Tuesday, September 30, 2025 at 03:04:16 PM
CAS
classification
:
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]
\begin{align*} 2 y^{\prime }+2 y {y^{\prime }}^{2}+\left (x +y^{2}\right ) y^{\prime \prime }&=a \end{align*}
✓ Maple. Time used: 0.017 (sec). Leaf size: 552
ode:=2*diff(y(x),x)+2*y(x)*diff(y(x),x)^2+(x+y(x)^2)*diff(diff(y(x),x),x) = a;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (6 a \,x^{2}-12 c_1 x +12 c_2 +2 \sqrt {9 a^{2} x^{4}+4 \left (-9 c_1 a +4\right ) x^{3}+36 \left (c_1^{2}+c_2 a \right ) x^{2}-72 c_1 x c_2 +36 c_2^{2}}\right )^{{2}/{3}}-4 x}{2 \left (6 a \,x^{2}-12 c_1 x +12 c_2 +2 \sqrt {9 a^{2} x^{4}+4 \left (-9 c_1 a +4\right ) x^{3}+36 \left (c_1^{2}+c_2 a \right ) x^{2}-72 c_1 x c_2 +36 c_2^{2}}\right )^{{1}/{3}}} \\
y &= -\frac {4 i \sqrt {3}\, x +i \left (6 a \,x^{2}-12 c_1 x +12 c_2 +2 \sqrt {9 a^{2} x^{4}+4 \left (-9 c_1 a +4\right ) x^{3}+36 \left (c_1^{2}+c_2 a \right ) x^{2}-72 c_1 x c_2 +36 c_2^{2}}\right )^{{2}/{3}} \sqrt {3}-4 x +\left (6 a \,x^{2}-12 c_1 x +12 c_2 +2 \sqrt {9 a^{2} x^{4}+4 \left (-9 c_1 a +4\right ) x^{3}+36 \left (c_1^{2}+c_2 a \right ) x^{2}-72 c_1 x c_2 +36 c_2^{2}}\right )^{{2}/{3}}}{4 \left (6 a \,x^{2}-12 c_1 x +12 c_2 +2 \sqrt {9 a^{2} x^{4}+4 \left (-9 c_1 a +4\right ) x^{3}+36 \left (c_1^{2}+c_2 a \right ) x^{2}-72 c_1 x c_2 +36 c_2^{2}}\right )^{{1}/{3}}} \\
y &= \frac {i \left (6 a \,x^{2}-12 c_1 x +12 c_2 +2 \sqrt {9 a^{2} x^{4}+4 \left (-9 c_1 a +4\right ) x^{3}+36 \left (c_1^{2}+c_2 a \right ) x^{2}-72 c_1 x c_2 +36 c_2^{2}}\right )^{{2}/{3}} \sqrt {3}+4 i \sqrt {3}\, x -\left (6 a \,x^{2}-12 c_1 x +12 c_2 +2 \sqrt {9 a^{2} x^{4}+4 \left (-9 c_1 a +4\right ) x^{3}+36 \left (c_1^{2}+c_2 a \right ) x^{2}-72 c_1 x c_2 +36 c_2^{2}}\right )^{{2}/{3}}+4 x}{4 \left (6 a \,x^{2}-12 c_1 x +12 c_2 +2 \sqrt {9 a^{2} x^{4}+4 \left (-9 c_1 a +4\right ) x^{3}+36 \left (c_1^{2}+c_2 a \right ) x^{2}-72 c_1 x c_2 +36 c_2^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 60.036 (sec). Leaf size: 441
ode=2*D[y[x],x] + 2*y[x]*D[y[x],x]^2 + (x + y[x]^2)*D[y[x],{x,2}] == a;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {-2\ 2^{2/3} x+\sqrt [3]{2} \left (3 a x^2+\sqrt {16 x^3+\left (3 a x^2+c_2 x+6 c_1\right ){}^2}+c_2 x+6 c_1\right ){}^{2/3}}{2 \sqrt [3]{3 a x^2+\sqrt {16 x^3+\left (3 a x^2+c_2 x+6 c_1\right ){}^2}+c_2 x+6 c_1}}\\ y(x)&\to \frac {i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (\sqrt {9 a^2 x^4+2 x^3 (8+3 a c_2)+x^2 \left (36 a c_1+c_2{}^2\right )+12 c_1 c_2 x+36 c_1{}^2}+3 a x^2+c_2 x+6 c_1\right ){}^{2/3}+2\ 2^{2/3} \left (1+i \sqrt {3}\right ) x}{4 \sqrt [3]{3 a x^2+\sqrt {16 x^3+\left (3 a x^2+c_2 x+6 c_1\right ){}^2}+c_2 x+6 c_1}}\\ y(x)&\to \frac {\sqrt [3]{2} \left (-1-i \sqrt {3}\right ) \left (\sqrt {9 a^2 x^4+2 x^3 (8+3 a c_2)+x^2 \left (36 a c_1+c_2{}^2\right )+12 c_1 c_2 x+36 c_1{}^2}+3 a x^2+c_2 x+6 c_1\right ){}^{2/3}+2\ 2^{2/3} \left (1-i \sqrt {3}\right ) x}{4 \sqrt [3]{3 a x^2+\sqrt {16 x^3+\left (3 a x^2+c_2 x+6 c_1\right ){}^2}+c_2 x+6 c_1}} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a + (x + y(x)**2)*Derivative(y(x), (x, 2)) + 2*y(x)*Derivative(y(x), x)**2 + 2*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -(sqrt(2*a*y(x) - 2*x*y(x)*Derivative(y(x), (x, 2)) - 2*y(x)**3*