Internal problem ID [6115]
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]
\[ \boxed {x y^{\prime }+y-x^{4} {y^{\prime }}^{2}=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 81
dsolve(y(x)+x*diff(y(x),x)=x^4*(diff(y(x),x))^2,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {1}{4 x^{2}} \\ y \left (x \right ) &= \frac {-c_{1} i-x}{c_{1}^{2} x} \\ y \left (x \right ) &= \frac {c_{1} i-x}{x \,c_{1}^{2}} \\ y \left (x \right ) &= \frac {c_{1} i-x}{x \,c_{1}^{2}} \\ y \left (x \right ) &= \frac {-c_{1} i-x}{c_{1}^{2} x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.5 (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} \text {arctanh}\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} \text {arctanh}\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*}