12.5.3 problem Example 3(a) (As Riccati)
Internal
problem
ID
[1627]
Book
:
Elementary
differential
equations
with
boundary
value
problems.
William
F.
Trench.
Brooks/Cole
2001
Section
:
Chapter
2,
First
order
equations.
Transformation
of
Nonlinear
Equations
into
Separable
Equations.
Section
2.4
Page
68
Problem
number
:
Example
3(a)
(As
Riccati)
Date
solved
:
Monday, January 27, 2025 at 05:15:10 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _Riccati]
\begin{align*} x^{2} y^{\prime }&=y^{2}+y x -x^{2} \end{align*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 12
dsolve(x^2*diff(y(x),x)=y(x)^2+x*y(x)-x^2,y(x), singsol=all)
\[
y = -\tanh \left (\ln \left (x \right )+c_1 \right ) x
\]
✓ Solution by Mathematica
Time used: 0.290 (sec). Leaf size: 298
DSolve[D[y[x],x]==y[x]^2+x*y[x]-x^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {5 \left (\sqrt {5}-1\right ) x \left (c_1 \operatorname {HermiteH}\left (\frac {1}{10} \left (-5+\sqrt {5}\right ),\frac {\sqrt [4]{5} x}{\sqrt {2}}\right )-\operatorname {Hypergeometric1F1}\left (\frac {5}{4}-\frac {1}{4 \sqrt {5}},\frac {3}{2},\frac {\sqrt {5} x^2}{2}\right )+\operatorname {Hypergeometric1F1}\left (\frac {1}{20} \left (5-\sqrt {5}\right ),\frac {1}{2},\frac {\sqrt {5} x^2}{2}\right )\right )-\sqrt {2} \sqrt [4]{5} \left (\sqrt {5}-5\right ) c_1 \operatorname {HermiteH}\left (\frac {1}{10} \left (-15+\sqrt {5}\right ),\frac {\sqrt [4]{5} x}{\sqrt {2}}\right )}{10 \left (\operatorname {Hypergeometric1F1}\left (\frac {1}{20} \left (5-\sqrt {5}\right ),\frac {1}{2},\frac {\sqrt {5} x^2}{2}\right )+c_1 \operatorname {HermiteH}\left (\frac {1}{10} \left (-5+\sqrt {5}\right ),\frac {\sqrt [4]{5} x}{\sqrt {2}}\right )\right )} \\
y(x)\to \frac {1}{2} \left (\sqrt {5}-1\right ) x-\frac {\left (\sqrt {5}-5\right ) \operatorname {HermiteH}\left (\frac {1}{10} \left (-15+\sqrt {5}\right ),\frac {\sqrt [4]{5} x}{\sqrt {2}}\right )}{\sqrt {2} 5^{3/4} \operatorname {HermiteH}\left (\frac {1}{10} \left (-5+\sqrt {5}\right ),\frac {\sqrt [4]{5} x}{\sqrt {2}}\right )} \\
\end{align*}