Internal problem ID [977]
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).
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _Riccati]
\[ \boxed {y^{\prime } x^{2}-y^{2}-y x=-x^{2}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 12
dsolve(x^2*diff(y(x),x)=y(x)^2+x*y(x)-x^2,y(x), singsol=all)
\[ y \left (x \right ) = -\tanh \left (\ln \left (x \right )+c_{1} \right ) x \]
✓ Solution by Mathematica
Time used: 0.298 (sec). Leaf size: 298
DSolve[y'[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*}