60.1.360 problem 368
Internal
problem
ID
[10374]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
368
Date
solved
:
Wednesday, March 05, 2025 at 10:41:18 AM
CAS
classification
:
[[_homogeneous, `class G`]]
\begin{align*} {y^{\prime }}^{2}+a y+b \,x^{2}&=0 \end{align*}
✗ Maple
ode:=diff(y(x),x)^2+a*y(x)+b*x^2 = 0;
dsolve(ode,y(x), singsol=all);
\[ \text {No solution found} \]
✓ Mathematica. Time used: 3.421 (sec). Leaf size: 795
ode=b*x^2 + a*y[x] + D[y[x],x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\int _1^{y(x)}\left (-\frac {\sqrt {-b x^2-a K[2]} x}{b x^4+a K[2] x^2+4 K[2]^2}-\int _1^x\left (\frac {a K[1]}{b K[1]^4+a K[2] K[1]^2+4 K[2]^2}+\frac {2 \sqrt {-b K[1]^2-a K[2]}}{b K[1]^4+a K[2] K[1]^2+4 K[2]^2}-\frac {a K[2]}{\sqrt {-b K[1]^2-a K[2]} \left (b K[1]^4+a K[2] K[1]^2+4 K[2]^2\right )}-\frac {\left (a K[1]^2+8 K[2]\right ) \left (b K[1]^3+a K[2] K[1]\right )}{\left (b K[1]^4+a K[2] K[1]^2+4 K[2]^2\right )^2}-\frac {2 K[2] \left (a K[1]^2+8 K[2]\right ) \sqrt {-b K[1]^2-a K[2]}}{\left (b K[1]^4+a K[2] K[1]^2+4 K[2]^2\right )^2}\right )dK[1]+\frac {2 K[2]}{b x^4+a K[2] x^2+4 K[2]^2}\right )dK[2]+\int _1^x\left (\frac {2 \sqrt {-b K[1]^2-a y(x)} y(x)}{b K[1]^4+a y(x) K[1]^2+4 y(x)^2}+\frac {b K[1]^3+a y(x) K[1]}{b K[1]^4+a y(x) K[1]^2+4 y(x)^2}\right )dK[1]&=c_1,y(x)\right ] \\
\text {Solve}\left [\int _1^{y(x)}\left (\frac {\sqrt {-b x^2-a K[4]} x}{b x^4+a K[4] x^2+4 K[4]^2}-\int _1^x\left (\frac {a K[3]}{b K[3]^4+a K[4] K[3]^2+4 K[4]^2}-\frac {2 \sqrt {-b K[3]^2-a K[4]}}{b K[3]^4+a K[4] K[3]^2+4 K[4]^2}+\frac {a K[4]}{\sqrt {-b K[3]^2-a K[4]} \left (b K[3]^4+a K[4] K[3]^2+4 K[4]^2\right )}-\frac {\left (a K[3]^2+8 K[4]\right ) \left (b K[3]^3+a K[4] K[3]\right )}{\left (b K[3]^4+a K[4] K[3]^2+4 K[4]^2\right )^2}+\frac {2 K[4] \left (a K[3]^2+8 K[4]\right ) \sqrt {-b K[3]^2-a K[4]}}{\left (b K[3]^4+a K[4] K[3]^2+4 K[4]^2\right )^2}\right )dK[3]+\frac {2 K[4]}{b x^4+a K[4] x^2+4 K[4]^2}\right )dK[4]+\int _1^x\left (\frac {b K[3]^3+a y(x) K[3]}{b K[3]^4+a y(x) K[3]^2+4 y(x)^2}-\frac {2 y(x) \sqrt {-b K[3]^2-a y(x)}}{b K[3]^4+a y(x) K[3]^2+4 y(x)^2}\right )dK[3]&=c_1,y(x)\right ] \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(a*y(x) + b*x**2 + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt(-a*y(x) - b*x**2) + Derivative(y(x), x) cannot be solved by the factorable group method