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]
\[ \boxed {y+y^{\prime } x -x^{4} {y^{\prime }}^{2}=0} \]
✓ Solution by Maple
Time used: 0.25 (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 = -\frac {1}{4 x^{2}} y = \frac {-c_{1} \left (2 i x -c_{1} \right )-c_{1}^{2}-2 x^{2}}{2 c_{1}^{2} x^{2}} y = \frac {-c_{1} \left (-2 i x -c_{1} \right )-c_{1}^{2}-2 x^{2}}{2 c_{1}^{2} x^{2}} y = \frac {c_{1} \left (2 i x +c_{1} \right )-2 x^{2}-c_{1}^{2}}{2 c_{1}^{2} x^{2}} y = \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: 0.498 (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*}