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−ax3+xy′(x)+xy(x)2−y(x)=0 ✓ Mathematica : cpu = 0.0203004 (sec), leaf count = 30
{{y(x)→axtanh(12a(2c1+x2))}}
✓ Maple : cpu = 0.056 (sec), leaf count = 22
{y(x)=tanh(x2+2_C12a)xa}
xy′+xy2−y−ax3=0 This is Riccati non-linear first order. But using the transformation y=xv it is transformed to easily solved ODEy′=v+xv′
Therefore the ODE becomes
x(v+xv′)+x(xv)2−xv−ax3=0xv+x2v′+x3v2−xv−ax3=0x2v′+x3v2−ax3=0v′+xv2−ax=0dvdx=x(a−v2)dva−v2=xdx
Integrating
1atanh−1(va)=x22+Ctanh−1(va)=a(x22+C)va=tanh(a(x22+C))v=atanh(a(x22+C))
Therefore
y=xv=xatanh(a(x22+C))
Verification
restart; ode:=x*diff(y(x),x)+x*y(x)^2-y(x)-a*x^3=0; my_solution:=x*sqrt(a)*tanh(sqrt(a)*(x^2/2+_C1)); odetest(y(x)=my_solution,ode); 0
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