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]
Solve \begin {gather*} \boxed {y^{\prime } x^{2}-y^{2}-y x +x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 12
dsolve(x^2*diff(y(x),x)=y(x)^2+x*y(x)-x^2,y(x), singsol=all)
\[ y \relax (x ) = -\tanh \left (\ln \relax (x )+c_{1}\right ) x \]
✓ Solution by Mathematica
Time used: 0.283 (sec). Leaf size: 244
DSolve[y'[x]==y[x]^2+x*y[x]-x^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{10} \left (5 \left (\sqrt {5}-1\right ) x+\frac {-5 \left (\sqrt {5}-1\right ) x \, _1F_1\left (\frac {5}{4}-\frac {1}{4 \sqrt {5}};\frac {3}{2};\frac {\sqrt {5} x^2}{2}\right )+2 \sqrt {5 \left (3 \sqrt {5}-5\right )} c_1 \text {HermiteH}\left (\frac {1}{10} \left (\sqrt {5}-15\right ),\frac {\sqrt [4]{5} x}{\sqrt {2}}\right )}{\, _1F_1\left (\frac {1}{20} \left (5-\sqrt {5}\right );\frac {1}{2};\frac {\sqrt {5} x^2}{2}\right )+c_1 \text {HermiteH}\left (\frac {1}{10} \left (\sqrt {5}-5\right ),\frac {\sqrt [4]{5} x}{\sqrt {2}}\right )}\right ) \\ y(x)\to \frac {\sqrt {\frac {3}{\sqrt {5}}-1} \text {HermiteH}\left (\frac {1}{10} \left (\sqrt {5}-15\right ),\frac {\sqrt [4]{5} x}{\sqrt {2}}\right )}{\text {HermiteH}\left (\frac {1}{10} \left (\sqrt {5}-5\right ),\frac {\sqrt [4]{5} x}{\sqrt {2}}\right )}+\frac {1}{2} \left (\sqrt {5}-1\right ) x \\ \end{align*}