83.19.4 problem 4

Internal problem ID [19232]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (B) at page 55
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 01:15:09 PM
CAS classification : [_quadrature]

\begin{align*} 3 {y^{\prime }}^{5}-y^{\prime } y+1&=0 \end{align*}

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 87

dsolve(3*diff(y(x),x)^5-diff(y(x),x)*y(x)+1=0,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {5 \operatorname {RootOf}\left (1+8 \textit {\_Z}^{5}+\left (-2 x +2 c_{1} \right ) \textit {\_Z}^{2}\right )^{3}+2 c_{1} -2 x}{2 \operatorname {RootOf}\left (1+8 \textit {\_Z}^{5}+\left (-2 x +2 c_{1} \right ) \textit {\_Z}^{2}\right ) \left (4 \operatorname {RootOf}\left (1+8 \textit {\_Z}^{5}+\left (-2 x +2 c_{1} \right ) \textit {\_Z}^{2}\right )^{3}+c_{1} -x \right )} \]

✓ Solution by Mathematica

Time used: 0.128 (sec). Leaf size: 176

DSolve[3*D[y[x],x]^5-D[y[x],x]*y[x]+1==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {Solve}\left [\int _1^{y(x)}\frac {1}{\text {Root}\left [3 \text {$\#$1}^5-K[1] \text {$\#$1}+1\&,1\right ]}dK[1]&=x+c_1,y(x)\right ] \\ \text {Solve}\left [\int _1^{y(x)}\frac {1}{\text {Root}\left [3 \text {$\#$1}^5-K[2] \text {$\#$1}+1\&,2\right ]}dK[2]&=x+c_1,y(x)\right ] \\ \text {Solve}\left [\int _1^{y(x)}\frac {1}{\text {Root}\left [3 \text {$\#$1}^5-K[3] \text {$\#$1}+1\&,3\right ]}dK[3]&=x+c_1,y(x)\right ] \\ \text {Solve}\left [\int _1^{y(x)}\frac {1}{\text {Root}\left [3 \text {$\#$1}^5-K[4] \text {$\#$1}+1\&,4\right ]}dK[4]&=x+c_1,y(x)\right ] \\ \text {Solve}\left [\int _1^{y(x)}\frac {1}{\text {Root}\left [3 \text {$\#$1}^5-K[5] \text {$\#$1}+1\&,5\right ]}dK[5]&=x+c_1,y(x)\right ] \\ \end{align*}