Internal problem ID [5343]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 10. Singular solutions, Extraneous loci. Supplemetary problems. Page
74
Problem number: 15.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class G`], _rational]
\[ \boxed {y+x y^{\prime }-x^{4} {y^{\prime }}^{2}=0} \]
✓ Solution by Maple
Time used: 0.062 (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}{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}{x \,c_{1}^{2}} \\ y \left (x \right ) &= \frac {-c_{1} i-x}{x \,c_{1}^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.498 (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*}