Internal problem ID [10192]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IV, differential equations of the first order and higher degree than the first. Article
25. Equations solvable for \(y\). Page 52
Problem number: Ex 5.
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: 0.219 (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 \left (x \right ) = -\frac {1}{4 x^{2}} \\ y \left (x \right ) = \frac {-c_{1} \left (2 i x -c_{1} \right )-c_{1}^{2}-2 x^{2}}{2 c_{1}^{2} x^{2}} \\ y \left (x \right ) = \frac {-c_{1} \left (-2 i x -c_{1} \right )-c_{1}^{2}-2 x^{2}}{2 c_{1}^{2} x^{2}} \\ y \left (x \right ) = \frac {c_{1} \left (2 i x +c_{1} \right )-2 x^{2}-c_{1}^{2}}{2 c_{1}^{2} x^{2}} \\ y \left (x \right ) = \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: 0.487 (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*}