Question

Hi I am really sorry I now people have already attempted to help me with this one already, however I am still not able to understand, I really have not done a lot of these equations and am struggling. I have put a little more information down this time, any help is much appreciated.

1- Find, in implicit form, the general solution of the differential equation:

Dy/dx = 2(e^x – e^-x) / y^2 (e^x + e^-x)^4 (y > 0).

2- Find the corresponding particular solution (in implicit form) that satisfies the initial; condition y = ½ when x = 0.

3- Find the explicit form of this particular solution.

4- What is the value of y given by this particular solution when x =1?

regards Claire

Answer 1

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.

Given:
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
u=e^x+e^(-x),
du=(e^x-e^(-x))dx
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:
y³=-2/(e^x+e^(-x))³+C

2. given y(0)=1/2 =>
(1/2)³=-2/(e^0+e^(-0))^3+C
1/8=-2/2³ + C
=>
C=1/8+1/4 = 3/8
The particular solution is therefore
y³=-2/(e^x+e^(-x))³+ 3/8

3. the explicit form is in the format
y=.....
=>
y=∛(-2/(e^x+e^(-x))³+ 3/8)
(y>0)

4. y(1)=? .... I'll leave that to you.

Do check my work.