Internal problem ID [5331]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 9. Equations of first order and higher degree. Supplemetary problems. Page
65
Problem number: 25.
ODE order: 1.
ODE degree: 5.
CAS Maple gives this as type [[_homogeneous, `class G`], _rational, _Clairaut]
\[ \boxed {x {y^{\prime }}^{5}-y {y^{\prime }}^{4}+\left (x^{2}+1\right ) {y^{\prime }}^{3}-2 x y {y^{\prime }}^{2}+\left (x +y^{2}\right ) y^{\prime }-y=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 37
dsolve(x*diff(y(x),x)^5-y(x)*diff(y(x),x)^4+(1+x^2)*diff(y(x),x)^3-2*x*y(x)*diff(y(x),x)^2+(x+y(x)^2)*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\begin{align*} y = c_{1}^{3}+c_{1} x y = c_{1} x^{\frac {3}{2}} y = c_{1} x +\frac {1}{c_{1}} y = c_{1} \sqrt {x} \end{align*}
✓ Solution by Mathematica
Time used: 0.042 (sec). Leaf size: 142
DSolve[x*y'[x]^5-y[x]*y'[x]^4+(1+x^2)*y'[x]^3-2*x*y[x]*y'[x]^2+(x+y[x]^2)*y'[x]-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 x+\frac {1}{c_1} y(x)\to c_1 \left (x+c_1{}^2\right ) y(x)\to \text {Indeterminate} y(x)\to -x-1 y(x)\to -2 \sqrt {x} y(x)\to 2 \sqrt {x} y(x)\to -\frac {2 i x^{3/2}}{3 \sqrt {3}} y(x)\to \frac {2 i x^{3/2}}{3 \sqrt {3}} y(x)\to x+1 y(x)\to -\sqrt {-(x-1)^2} y(x)\to \sqrt {-(x-1)^2} \end{align*}