40.5.9 problem 25

Internal problem ID [6674]
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
Date solved : Monday, January 27, 2025 at 02:18:58 PM
CAS classification : [[_homogeneous, `class G`], _rational, _Clairaut]

\begin{align*} 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 \end{align*}

✓ Solution by Maple

Time used: 0.059 (sec). Leaf size: 63

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 &= \frac {2 \sqrt {3}\, \left (-x \right )^{{3}/{2}}}{9} \\ y &= -\frac {2 \sqrt {3}\, \left (-x \right )^{{3}/{2}}}{9} \\ y &= c_1 \left (c_1^{2}+x \right ) \\ y &= -2 \sqrt {x} \\ y &= 2 \sqrt {x} \\ y &= c_1 x +\frac {1}{c_1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.044 (sec). Leaf size: 142

DSolve[x*D[y[x],x]^5-y[x]*D[y[x],x]^4+(1+x^2)*D[y[x],x]^3-2*x*y[x]*D[y[x],x]^2+(x+y[x]^2)*D[y[x],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*}