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
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.
Hi Claire! I'll do my best to help you understand and solve the differential equation step by step.
1. To find the general solution of the given differential equation, we need to rearrange it to separate the variables x and y. Let's start by multiplying both sides by y^2 and (e^x + e^-x)^4:
y^2 * dy/dx = 2(e^x – e^-x) / (e^x + e^-x)^4
Now, let's simplify the right-hand side by expanding (e^x + e^-x)^4 to (e^4x + 4e^2x + 6 + 4e^-2x + e^-4x):
y^2 * dy/dx = 2(e^x – e^-x) / (e^4x + 4e^2x + 6 + 4e^-2x + e^-4x)
Next, let's move y^2 to the right-hand side by dividing both sides by y^2:
(dy/dx) / y^2 = 2(e^x – e^-x) / (e^4x + 4e^2x + 6 + 4e^-2x + e^-4x)
Now, we can rewrite dy/dx as d/dx(y):
d/dx(y) / y^2 = 2(e^x – e^-x) / (e^4x + 4e^2x + 6 + 4e^-2x + e^-4x)
To make further progress, let's rewrite d/dx(y) / y^2 as (1/y^2) * d/dx(y):
(1/y^2) * d/dx(y) = 2(e^x – e^-x) / (e^4x + 4e^2x + 6 + 4e^-2x + e^-4x)
Now, we can separate the variables by multiplying both sides by dx:
(1/y^2) * d/dx(y) * dx = 2(e^x – e^-x) / (e^4x + 4e^2x + 6 + 4e^-2x + e^-4x) * dx
Next, let's integrate both sides with respect to x:
∫(1/y^2) * d/dx(y) * dx = ∫[2(e^x – e^-x) / (e^4x + 4e^2x + 6 + 4e^-2x + e^-4x)] * dx
The left-hand side can be simplified as:
∫(1/y^2) * d/dx(y) * dx = ∫d/dx(1/y) * dx
Using the chain rule, d/dx(1/y) = -1/y^2 * dy/dx:
∫d/dx(1/y) * dx = -∫(1/y^2) * dy/dx * dx
Now we can substitute -dy/dx for (1/y^2) * d/dx(y):
-∫(1/y^2) * dy/dx * dx = -∫-dy/y^2
Simplifying further:
∫-dy/y^2 = ∫dy/y^2
Integrating both sides:
-ln|y| = x + C
Where C is a constant of integration.
2. To find the particular solution that satisfies the initial condition y = 1/2 when x = 0, we can substitute these values into the equation:
-ln|1/2| = 0 + C
Simplifying:
ln(2) = C
So the particular solution that satisfies the initial condition is:
-ln|y| = x + ln(2)
3. To find the explicit form of this particular solution, we can solve for y by taking the exponential function of both sides:
e^(-ln|y|) = e^(x + ln(2))
Simplifying:
y^(-1) = e^(x) * 2
Taking the reciprocal:
y = 1 / (e^x * 2)
4. To find the value of y given by this particular solution when x = 1, substitute x = 1 into the explicit form:
y = 1 / (e^1 * 2)
Simplifying:
y = 1 / (e^1 * 2)
y = 1 / (2.71828 * 2)
y ≈ 0.18394
Thus, when x = 1, y ≈ 0.18394.
I hope this explanation helps you understand the steps involved in solving the differential equation. If you have any further questions, feel free to ask!