6.5.3 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 : Tuesday, September 30, 2025 at 04:40:49 AM
CAS classification : [[_homogeneous, `class A`], _rational, _Riccati]

\begin{align*} x^{2} y^{\prime }&=y^{2}+x y-x^{2} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 12

ode:=x^2*diff(y(x),x) = y(x)^2+x*y(x)-x^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\tanh \left (\ln \left (x \right )+c_1 \right ) x \]

✓ Mathematica. Time used: 0.164 (sec). Leaf size: 298

ode=D[y[x],x]==y[x]^2+x*y[x]-x^2; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 0.180 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), x) + x**2 - x*y(x) - y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {x \left (C_{1} - x^{2} - 1\right )}{C_{1} + x^{2} - 1} \]