56.1.69 problem 69
Internal
problem
ID
[8781]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
1.0
Problem
number
:
69
Date
solved
:
Wednesday, March 05, 2025 at 06:48:21 AM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} a y^{2} y^{\prime \prime }+b y^{2}&=c \end{align*}
✓ Maple. Time used: 0.032 (sec). Leaf size: 76
ode:=a*y(x)^2*diff(diff(y(x),x),x)+b*y(x)^2 = c;
dsolve(ode,y(x), singsol=all);
\begin{align*}
a \left (\int _{}^{y}\frac {\textit {\_a}}{\sqrt {\textit {\_a} a \left (c_{1} \textit {\_a} a -2 b \,\textit {\_a}^{2}-2 c \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
-a \left (\int _{}^{y}\frac {\textit {\_a}}{\sqrt {\textit {\_a} a \left (c_{1} \textit {\_a} a -2 b \,\textit {\_a}^{2}-2 c \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.85 (sec). Leaf size: 346
ode=a*y[x]^2*D[y[x],{x,2}]+b*y[x]^2==c;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [-\frac {\left (\sqrt {-16 b c+a^2 c_1{}^2}-a c_1\right ) \left (\sqrt {-16 b c+a^2 c_1{}^2}+a c_1\right ){}^2 \left (1+\frac {4 b y(x)}{\sqrt {-16 b c+a^2 c_1{}^2}-a c_1}\right ) \left (1-\frac {4 b y(x)}{\sqrt {-16 b c+a^2 c_1{}^2}+a c_1}\right ) \left (E\left (i \text {arcsinh}\left (2 \sqrt {\frac {b}{\sqrt {a^2 c_1{}^2-16 b c}-a c_1}} \sqrt {y(x)}\right )|\frac {a c_1-\sqrt {a^2 c_1{}^2-16 b c}}{a c_1+\sqrt {a^2 c_1{}^2-16 b c}}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (2 \sqrt {\frac {b}{\sqrt {a^2 c_1{}^2-16 b c}-a c_1}} \sqrt {y(x)}\right ),\frac {a c_1-\sqrt {a^2 c_1{}^2-16 b c}}{a c_1+\sqrt {a^2 c_1{}^2-16 b c}}\right )\right ){}^2}{16 b^3 y(x) \left (-\frac {2 \left (b y(x)^2+c\right )}{a y(x)}+c_1\right )}=(x+c_2){}^2,y(x)\right ]
\]
✓ Sympy. Time used: 95.329 (sec). Leaf size: 48
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(a*y(x)**2*Derivative(y(x), (x, 2)) + b*y(x)**2 - c,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {C_{1} - \frac {2 \left (u b + \frac {c}{u}\right )}{a}}}\, du = C_{2} + x, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {C_{1} - \frac {2 \left (u b + \frac {c}{u}\right )}{a}}}\, du = C_{2} - x\right ]
\]