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−y(x)2f′(x)g(x)+g′(x)f(x)+y′(x)=0 ✓ Mathematica : cpu = 0.48339 (sec), leaf count = 160
Solve[∫1y(x)(1(g(x)+f(x)K[2])2−∫1x(2(f(K[1])K[2]2f′(K[1])−g(K[1])g′(K[1]))g(K[1])(g(K[1])+f(K[1])K[2])3−2K[2]f′(K[1])g(K[1])(g(K[1])+f(K[1])K[2])2)dK[1])dK[2]+∫1x−f(K[1])y(x)2f′(K[1])−g(K[1])g′(K[1])f(K[1])g(K[1])(g(K[1])+f(K[1])y(x))2dK[1]=c1,y(x)] ✓ Maple : cpu = 0.457 (sec), leaf count = 58
{y(x)=−c1f(x)g(x)−f(x)g(x)(∫ddxf(x)f(x)2g(x)dx)−1(c1+∫ddxf(x)f(x)2g(x)dx)f(x)2}
−f′gy2+g′f+y′=0y′=−g′f+f′gy2(1)=P(x)+Q(x)y+R(x)y2
This is Ricatti first order non-linear ODE. P(x)=−g′f,Q(x)=0,R(x)=f′g.
To do.
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