If dy/dx = (1 + x)/(xy), x>0, and y=-4 when x=1, then when x=3, y= what?
It's -4.711393061, to find out what sign you need, you need to consider the sign of dy/dx. You can see that because y = -4 at x = 1, dy/dx is negative, so at that point y(x) is negative and decreasing. It can't become positive, because that would require there to be a point where dy/dx = 0 between x = 1 and x = 3( assuming y is continuous), but x + 1/x doesn't become zero on that interval.
After integrating and solving, I got approximately 4.711393061. Does this seem right?
y dy = (1+x)/x dx = (1 + 1/x) dx
Integrate both sides
To find the value of y when x = 3, we can use the given differential equation and the initial condition y = -4 when x = 1. Let's solve the differential equation step by step:
Step 1: Separate the variables.
dy/(1 + x) = dx/(x * y)
Step 2: Integrate both sides of the equation.
∫ (1 + x)^(-1) dy = ∫ (x * y)^(-1) dx
Step 3: Integrate the left side of the equation by using the reverse power rule:
ln|1 + x| = ln|x * y| + C1
Step 4: Simplify the right side by using the properties of logarithms:
ln|1 + x| = ln|x| + ln|y| + C1
Step 5: Combine the logarithms on the right side:
ln|1 + x| - ln|x| = ln|y| + C1
ln((1 + x)/x) = ln|y| + C1
Step 6: Exponentiate both sides of the equation:
(1 + x)/x = K * |y|
Here, K is a constant obtained by exponentiating the constant C1.
Step 7: Rearrange the equation to isolate y:
|y| = x/(1 + x)
Since x > 0 in the given conditions, we can drop the absolute value signs.
Step 8: Substitute the initial condition y = -4 when x = 1:
|-4| = 1/(1 + 1)
4 = 1/2
This means that there is a contradiction with the given initial condition, suggesting that the initial condition might be incorrect, or the differential equation might not have a unique solution.
Therefore, we cannot determine the value of y when x = 3 with the given information.
Well, let's put on our funny hats and solve this equation!
First, let's separate and integrate both sides with respect to x. We have:
∫(1 + x)/(xy) dx = ∫dy/y
Integrating the left side gives us:
ln(xy) = ∫dy/y
ln(xy) = ln|y| + C
Now, let's use the initial condition y = -4 when x = 1 to find C. Plugging these values into our equation, we get:
ln(1*(-4)) = ln|-4| + C
ln|-4| = ln(4) because the natural log of a negative number isn't defined in the real domain.
ln(-4) = ln(4) + C
Now, let's find C:
C = ln(-4) - ln(4)
C = ln(-1) + ln(4) - ln(4)
C = ln(-1) + ln(e^2) - ln(e^2)
C = ln(-1)
And since we know ln(-1) = πi (some imaginary number), we can substitute it into our equation:
ln(xy) = ln|y| + πi
Now, let's convert our equation back to exponential form:
xy = y * e^πi
Notice that e^πi = -1 (thanks, Euler!). Simplifying our equation:
xy = -y
Now we can substitute x = 3 into our equation to find y when x = 3:
3y = -y
2y = 0
Dividing by 2:
y = 0
So, when x = 3, y = 0. Voilà!