Wednesday

October 22, 2014

October 22, 2014

Posted by **Claire** on Thursday, February 17, 2011 at 11:38am.

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

- Maths-differential equation -
**MathMate**, Thursday, February 17, 2011 at 1:27pmThis 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.

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