Solve the seperable differential equation 8yy�Œ =x. Use the following initial condition: y(8)=4.
Express x^2 in terms of y.
8yy' = x
8y dy = x dx
4y^2 = 1/2 x^2 + c
4(16) = 1/2 (64) + c
c = 32
4y^2 = 1/2 x^2 + 32
8y^2 - x^2 = 64
y^2/8 - x^2/64 = 1
To solve the separable differential equation 8yy' = x, we can rearrange the equation as:
8y dy = x dx
Now, we can integrate both sides of the equation:
∫8y dy = ∫x dx
This gives us:
4y^2 = 0.5x^2 + C
Next, we use the initial condition y(8) = 4 to find the value of the constant C. Substituting x = 8 and y = 4 into the equation, we have:
4(4)^2 = 0.5(8)^2 + C
64 = 32 + C
C = 32
Substituting the value of C back into the equation, we have:
4y^2 = 0.5x^2 + 32
To express x^2 in terms of y, we rearrange the equation as:
0.5x^2 = 4y^2 - 32
Multiplying both sides by 2, we get:
x^2 = 8y^2 - 64
Therefore, x^2 can be expressed in terms of y as 8y^2 - 64.
To solve the separable differential equation, we need to separate the variables and integrate.
Given the differential equation:
8yy' = x
To separate the variables, we can divide both sides by 8y:
y' = x/(8y)
Now, we can separate the variables by moving the y terms to one side and the x terms to the other side:
dy/y = x/(8y) dx
Next, we integrate both sides with respect to their respective variables:
∫(dy/y) = ∫(x/(8y)) dx
The left side is the natural logarithm of y, and the right side can be simplified:
ln|y| = (1/8)∫(1/y) dx
Now, let's integrate the right side:
ln|y| = (1/8)∫(1/y) dx = (1/8)ln|y| + C1
Where C1 is the constant of integration.
Now, rearrange the equation to isolate ln|y| on one side:
ln|y| - (1/8)ln|y| = C1
Combine the logarithms:
(7/8)ln|y| = C1
Raise both sides to the power of e to eliminate the logarithm:
e^(7/8)ln|y| = e^C1
Since e^C1 is just another constant C2, we have:
|y|^(7/8) = C2
Note that |y| denotes the absolute value of y.
Now, we can solve for the constant by using the initial condition y(8) = 4:
|4|^(7/8) = C2
Taking the absolute value of 4 and raising it to the power of 7/8, we get:
4^(7/8) = C2
Simplifying further, we have:
2^(7/4) = C2
So, the constant C2 is equal to 2^(7/4).
Now, let's express x^2 in terms of y. We can go back to the original differential equation:
8yy' = x
Now, divide both sides by 8y and multiply by dx:
y' = x/(8y) dx
Integrate both sides:
∫dy = ∫(x/(8y)) dx
The left side is just y, and the right side can be simplified:
y = (1/8)∫(x/y) dx
Now, rewrite the integral:
y = (1/8)∫(1/y)x dx
Using the fact that x^2 = (x)(x), we can rewrite the integral as:
y = (1/8)∫(1/y)x dx = (1/8)∫(x)(x) (1/y) dx
Now, we can express x^2 in terms of y by substituting (1/8)y for (1/y):
y = (1/8)∫(x)(x) (1/y) dx = (1/8)∫(x^2)(1/(1/8)y) dx
Simplifying the denominator:
y = (1/8)∫(x^2)(8/y) dx
The y terms cancel out, leaving:
y = ∫x^2 dx
Integrating x^2, we get:
y = (1/3)x^3 + C3
Where C3 is the constant of integration.
So, x^2 can be expressed in terms of y as:
x^2 = 3y - C3