60.2.341 problem 918

Internal problem ID [10928]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 918
Date solved : Monday, January 27, 2025 at 10:25:10 PM
CAS classification : [_rational]

\begin{align*} y^{\prime }&=\frac {2 y^{8}}{y^{5}+2 y^{6}+2 y^{2}+16 x y^{4}+32 y^{6} x^{2}+2+24 x y^{2}+96 x^{2} y^{4}+128 x^{3} y^{6}} \end{align*}

✓ Solution by Maple

Time used: 0.060 (sec). Leaf size: 41

dsolve(diff(y(x),x) = 2*y(x)^8/(y(x)^5+2*y(x)^6+2*y(x)^2+16*x*y(x)^4+32*y(x)^6*x^2+2+24*x*y(x)^2+96*x^2*y(x)^4+128*x^3*y(x)^6),y(x), singsol=all)
 

\[ x -\operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {1}{64 \textit {\_a}^{3}+16 \textit {\_a}^{2}+1}d \textit {\_a} \right ) y+c_{1} y+1\right )+\frac {1}{4 y^{2}} = 0 \]

✓ Solution by Mathematica

Time used: 0.254 (sec). Leaf size: 344

DSolve[D[y[x],x] == (2*y[x]^8)/(2 + 2*y[x]^2 + 24*x*y[x]^2 + 16*x*y[x]^4 + 96*x^2*y[x]^4 + y[x]^5 + 2*y[x]^6 + 32*x^2*y[x]^6 + 128*x^3*y[x]^6),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {K[2]^3}{2 \left (64 x^3 K[2]^6+16 x^2 K[2]^6+K[2]^6+48 x^2 K[2]^4+8 x K[2]^4+12 x K[2]^2+K[2]^2+1\right )}-\int _1^x\left (\frac {K[2]^6 \left (384 K[1]^3 K[2]^5+96 K[1]^2 K[2]^5+6 K[2]^5+192 K[1]^2 K[2]^3+32 K[1] K[2]^3+24 K[1] K[2]+2 K[2]\right )}{\left (64 K[1]^3 K[2]^6+16 K[1]^2 K[2]^6+K[2]^6+48 K[1]^2 K[2]^4+8 K[1] K[2]^4+12 K[1] K[2]^2+K[2]^2+1\right )^2}-\frac {6 K[2]^5}{64 K[1]^3 K[2]^6+16 K[1]^2 K[2]^6+K[2]^6+48 K[1]^2 K[2]^4+8 K[1] K[2]^4+12 K[1] K[2]^2+K[2]^2+1}\right )dK[1]+\frac {1}{K[2]^2}\right )dK[2]+\int _1^x-\frac {y(x)^6}{64 K[1]^3 y(x)^6+16 K[1]^2 y(x)^6+y(x)^6+48 K[1]^2 y(x)^4+8 K[1] y(x)^4+12 K[1] y(x)^2+y(x)^2+1}dK[1]=c_1,y(x)\right ] \]