60.2.301 problem 878

Internal problem ID [10888]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 878
Date solved : Tuesday, January 28, 2025 at 05:29:08 PM
CAS classification : [_rational]

\begin{align*} y^{\prime }&=\frac {1+y^{4}-8 a x y^{2}+16 a^{2} x^{2}+y^{6}-12 y^{4} a x +48 y^{2} a^{2} x^{2}-64 a^{3} x^{3}}{y} \end{align*}

✓ Solution by Maple

Time used: 18.809 (sec). Leaf size: 73

dsolve(diff(y(x),x) = (1+y(x)^4-8*a*x*y(x)^2+16*a^2*x^2+y(x)^6-12*y(x)^4*a*x+48*y(x)^2*a^2*x^2-64*a^3*x^3)/y(x),y(x), singsol=all)
 

\[ -\int _{\textit {\_b}}^{y}\frac {\textit {\_a}}{\textit {\_a}^{6}-12 \textit {\_a}^{4} a x +48 \textit {\_a}^{2} a^{2} x^{2}-64 a^{3} x^{3}+\textit {\_a}^{4}-8 \textit {\_a}^{2} a x +16 a^{2} x^{2}-2 a +1}d \textit {\_a} +x -c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.322 (sec). Leaf size: 286

DSolve[D[y[x],x] == (1 + 16*a^2*x^2 - 64*a^3*x^3 - 8*a*x*y[x]^2 + 48*a^2*x^2*y[x]^2 + y[x]^4 - 12*a*x*y[x]^4 + y[x]^6)/y[x],y[x],x,IncludeSingularSolutions -> True]
 

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