2.102   ODE No. 102

  1. Problem in Latex
  2. Mathematica input
  3. Maple input

ax3+xy(x)+xy(x)2y(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}

Hand solution

xy+xy2yax3=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)2xvax3=0xv+x2v+x3v2xvax3=0x2v+x3v2ax3=0v+xv2ax=0dvdx=x(av2)dvav2=xdx

Integrating

1atanh1(va)=x22+Ctanh1(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