32.10.16 problem Exercise 35.16, page 504
Internal
problem
ID
[6010]
Book
:
Ordinary
Differential
Equations,
By
Tenenbaum
and
Pollard.
Dover,
NY
1963
Section
:
Chapter
8.
Special
second
order
equations.
Lesson
35.
Independent
variable
x
absent
Problem
number
:
Exercise
35.16,
page
504
Date
solved
:
Wednesday, March 05, 2025 at 12:03:53 AM
CAS
classification
:
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
\begin{align*} \left (y+1\right ) y^{\prime \prime }&=3 {y^{\prime }}^{2} \end{align*}
With initial conditions
\begin{align*} y \left (1\right )&=0\\ y^{\prime }\left (1\right )&=-{\frac {1}{2}} \end{align*}
✓ Maple. Time used: 0.333 (sec). Leaf size: 15
ode:=(1+y(x))*diff(diff(y(x),x),x) = 3*diff(y(x),x)^2;
ic:=y(1) = 0, D(y)(1) = -1/2;
dsolve([ode,ic],y(x), singsol=all);
\[
y = \frac {-x +\sqrt {x}}{x}
\]
✓ Mathematica. Time used: 1.312 (sec). Leaf size: 572
ode=(y[x]+1)*D[y[x],{x,2}]==3*(D[y[x],x])^2;
ic={y[1]==0,Derivative[1][y][0] ==-1/2};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {6 \left (\left (-12+3\ 2^{2/3} \sqrt [3]{27-3 \sqrt {69}}-\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+3\ 2^{2/3} \sqrt [3]{3 \left (9+\sqrt {69}\right )}-\sqrt [3]{2} \left (3 \left (9+\sqrt {69}\right )\right )^{2/3}\right ) x+3 \sqrt {2} \sqrt {\left (12-3\ 2^{2/3} \sqrt [3]{27-3 \sqrt {69}}+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}-3\ 2^{2/3} \sqrt [3]{3 \left (9+\sqrt {69}\right )}+\sqrt [3]{2} \left (3 \left (9+\sqrt {69}\right )\right )^{2/3}\right ) x-\sqrt [3]{2} \left (3 \left (9+\sqrt {69}\right )\right )^{2/3}+2\ 2^{2/3} \sqrt [3]{3 \left (9+\sqrt {69}\right )}+\sqrt [3]{9-\sqrt {69}} \left (9+\sqrt {69}\right )^{2/3}+\left (9-\sqrt {69}\right )^{2/3} \sqrt [3]{9+\sqrt {69}}-\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+2\ 2^{2/3} \sqrt [3]{27-3 \sqrt {69}}+6}+\sqrt [3]{2} \left (3 \left (9+\sqrt {69}\right )\right )^{2/3}-2\ 2^{2/3} \sqrt [3]{3 \left (9+\sqrt {69}\right )}-\sqrt [3]{9-\sqrt {69}} \left (9+\sqrt {69}\right )^{2/3}-\left (9-\sqrt {69}\right )^{2/3} \sqrt [3]{9+\sqrt {69}}+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}-2\ 2^{2/3} \sqrt [3]{27-3 \sqrt {69}}-6\right )}{\left (12-3\ 2^{2/3} \sqrt [3]{27-3 \sqrt {69}}+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}-3\ 2^{2/3} \sqrt [3]{3 \left (9+\sqrt {69}\right )}+\sqrt [3]{2} \left (3 \left (9+\sqrt {69}\right )\right )^{2/3}\right ) \left (6 x+2^{2/3} \sqrt [3]{3} \left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right )\right )}
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((y(x) + 1)*Derivative(y(x), (x, 2)) - 3*Derivative(y(x), x)**2,0)
ics = {y(1): 0, Subs(Derivative(y(x), x), x, 1): -1/2}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt(3)*sqrt((y(x) + 1)*Derivative(y(x), (x, 2)))/3 + Derivative(y(x), x) cannot be solved by the factorable group method