This differential equation is separable, meaning that it can be transformed into two integrals, one involving only x, and the other, only y.
It can then be integrated to get the solution in terms of an integration constant. The initial conditions can be used to determine the constant.
Dy/dx = 2(e^x – e^-x) / y^2 (e^x + e^-x)^4 (y > 0)
transpose x and y terms to give:
y² dy = 2(e^x – e^-x) dx / (e^x + e^-x)^4
Note how the numerator is related to the derivative of the denominator, so we can use a substitution
so the equation becomes:
y² dy = 2du /u^4
Integrate both sides:
y³/3 = -(2/3)u^(-3)+C'
where C is an integration constant.
The general solution is therefore:
2. given y(0)=1/2 =>
1/8=-2/2³ + C
C=1/8+1/4 = 3/8
The particular solution is therefore
3. the explicit form is in the format
4. y(1)=? .... I'll leave that to you.
Do check my work.
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