Internal problem ID [5362]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.2 THE NATURE OF SOLUTIONS.
Page 9
Problem number: 1(L).
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class G], _rational]
Solve \begin {gather*} \boxed {x y^{\prime }+y-x^{4} \left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 1.052 (sec). Leaf size: 135
dsolve(y(x)+x*diff(y(x),x)=x^4*(diff(y(x),x))^2,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {1}{4 x^{2}} \\ y \relax (x ) = \frac {-c_{1}^{2}-c_{1} \left (2 i x -c_{1}\right )-2 x^{2}}{2 x^{2} c_{1}^{2}} \\ y \relax (x ) = \frac {-c_{1}^{2}-c_{1} \left (-2 i x -c_{1}\right )-2 x^{2}}{2 x^{2} c_{1}^{2}} \\ y \relax (x ) = \frac {c_{1} \left (2 i x +c_{1}\right )-2 x^{2}-c_{1}^{2}}{2 c_{1}^{2} x^{2}} \\ y \relax (x ) = \frac {c_{1} \left (-2 i x +c_{1}\right )-2 x^{2}-c_{1}^{2}}{2 c_{1}^{2} x^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.581 (sec). Leaf size: 123
DSolve[y[x]+x*y'[x]==x^4*(y'[x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [-\frac {x \sqrt {4 x^2 y(x)+1} \tanh ^{-1}\left (\sqrt {4 x^2 y(x)+1}\right )}{\sqrt {4 x^4 y(x)+x^2}}-\frac {1}{2} \log (y(x))=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {x \sqrt {4 x^2 y(x)+1} \tanh ^{-1}\left (\sqrt {4 x^2 y(x)+1}\right )}{\sqrt {4 x^4 y(x)+x^2}}-\frac {1}{2} \log (y(x))=c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}