54.1.360 problem 368
Internal
problem
ID
[11674]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
368
Date
solved
:
Tuesday, September 30, 2025 at 10:06:12 PM
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: 0.88 (sec). Leaf size: 581
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 [\text {RootSum}\left [\text {$\#$1}^4-\text {$\#$1}^3 a+2 \text {$\#$1}^2 b+\text {$\#$1} a b+b^2\&,\frac {2 \text {$\#$1}^3 \log \left (\text {$\#$1} x-\sqrt {-a y(x)-b x^2}+\sqrt {-a y(x)}\right )-2 \text {$\#$1}^3 \log (x)-\text {$\#$1}^2 a \log \left (\text {$\#$1} x-\sqrt {-a y(x)-b x^2}+\sqrt {-a y(x)}\right )+\text {$\#$1}^2 a \log (x)+2 \text {$\#$1} b \log \left (\text {$\#$1} x-\sqrt {-a y(x)-b x^2}+\sqrt {-a y(x)}\right )+a b \log \left (\text {$\#$1} x-\sqrt {-a y(x)-b x^2}+\sqrt {-a y(x)}\right )-2 \text {$\#$1} b \log (x)-a b \log (x)}{4 \text {$\#$1}^3-3 \text {$\#$1}^2 a+4 \text {$\#$1} b+a b}\&\right ]-\log \left (\sqrt {-a y(x)} \sqrt {-a y(x)-b x^2}+a y(x)\right )+\frac {1}{2} \log (y(x))+2 \log (x)=c_1,y(x)\right ]\\ \text {Solve}\left [\text {RootSum}\left [\text {$\#$1}^4+\text {$\#$1}^3 a+2 \text {$\#$1}^2 b-\text {$\#$1} a b+b^2\&,\frac {-2 \text {$\#$1}^3 \log \left (\text {$\#$1} x-\sqrt {-a y(x)-b x^2}+\sqrt {-a y(x)}\right )+2 \text {$\#$1}^3 \log (x)-\text {$\#$1}^2 a \log \left (\text {$\#$1} x-\sqrt {-a y(x)-b x^2}+\sqrt {-a y(x)}\right )+\text {$\#$1}^2 a \log (x)-2 \text {$\#$1} b \log \left (\text {$\#$1} x-\sqrt {-a y(x)-b x^2}+\sqrt {-a y(x)}\right )+a b \log \left (\text {$\#$1} x-\sqrt {-a y(x)-b x^2}+\sqrt {-a y(x)}\right )+2 \text {$\#$1} b \log (x)-a b \log (x)}{-4 \text {$\#$1}^3-3 \text {$\#$1}^2 a-4 \text {$\#$1} b+a b}\&\right ]-\log \left (\sqrt {-a y(x)} \sqrt {-a y(x)-b x^2}+a y(x)\right )+\frac {1}{2} \log (y(x))+2 \log (x)=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 b