Friday

December 19, 2014

December 19, 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.

**Answer this Question**

**Related Questions**

English - Sorry, for asking you the same thing again, but I really don't know if...

Dear Jiskha - Dear Jiskha, I really need help with some things and I am able to ...

English - I'm really sorry. The last two sentences in my last post shouldn't be ...

math - How many two digit numbers are there in base 5. I really don't understand...

English - Writeacher, I really hope you can have a look at my last examples. I ...

Advice on IB subjects - Hey there all! Im desperate for some advice on my ...

math - I need help with this question. Write the equation of the line which ...

English - I urgently need you to check these sentences on the present perfect. ...

math-moneyfractions - i have some math questions that i really ,really,need help...

history - I am doing a history project about Martin Ward's eviction in Loughrea ...