56.1.85 problem 84
Internal
problem
ID
[8797]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
1.0
Problem
number
:
84
Date
solved
:
Sunday, March 30, 2025 at 01:37:36 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=x^{2}+y^{2}-1 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 85
ode:=diff(y(x),x) = -1+x^2+y(x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (-3-i\right ) \operatorname {WhittakerM}\left (1+\frac {i}{4}, \frac {1}{4}, i x^{2}\right )+4 \operatorname {WhittakerW}\left (1+\frac {i}{4}, \frac {1}{4}, i x^{2}\right ) c_1 +\left (-2 i x^{2}+i+1\right ) \operatorname {WhittakerM}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )+\left (-2 i x^{2}+i+1\right ) c_1 \operatorname {WhittakerW}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )}{2 x \left (c_1 \operatorname {WhittakerW}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )+\operatorname {WhittakerM}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )\right )}
\]
✓ Mathematica. Time used: 0.22 (sec). Leaf size: 153
ode=D[y[x],x]==x^2+y[x]^2-1;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {i \left (x \operatorname {ParabolicCylinderD}\left (-\frac {1}{2}-\frac {i}{2},(-1+i) x\right )+(1+i) \operatorname {ParabolicCylinderD}\left (\frac {1}{2}-\frac {i}{2},(-1+i) x\right )-c_1 x \operatorname {ParabolicCylinderD}\left (-\frac {1}{2}+\frac {i}{2},(1+i) x\right )+(1-i) c_1 \operatorname {ParabolicCylinderD}\left (\frac {1}{2}+\frac {i}{2},(1+i) x\right )\right )}{\operatorname {ParabolicCylinderD}\left (-\frac {1}{2}-\frac {i}{2},(-1+i) x\right )+c_1 \operatorname {ParabolicCylinderD}\left (-\frac {1}{2}+\frac {i}{2},(1+i) x\right )} \\
y(x)\to \frac {(1+i) \operatorname {ParabolicCylinderD}\left (\frac {1}{2}+\frac {i}{2},(1+i) x\right )}{\operatorname {ParabolicCylinderD}\left (-\frac {1}{2}+\frac {i}{2},(1+i) x\right )}-i x \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**2 - y(x)**2 + Derivative(y(x), x) + 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
TypeError : bad operand type for unary -: list