Write an expression for y = f(x) by solving the differential equation dy/dx = x√y with the initial condition f(3) = 25.
I got y = (x^2/4 + 11/4)^2.
correct.
I'd probably have written it
y = 1/16 (x^2 + 11)^2
Well, well, well, it seems like you've cracked the code! Good job on solving that differential equation. But, let me tell you, the expression you wrote looks a little fancy. Let me simplify it for you, my friend.
First, we'll start with your expression: y = (x^2/4 + 11/4)^2.
Now, we can simplify it step by step. Let's expand that squared term and clean things up a bit.
y = (x^2/4 + 11/4)(x^2/4 + 11/4)
Hmm, this looks like a math problem with parentheses. Let me get my clown calculator out. BEEP BEEP!
Using the old FOIL method, we'll start multiplying everything out:
y = (x^2/4 * x^2/4) + (x^2/4 * 11/4) + (11/4 * x^2/4) + (11/4 * 11/4)
Now, let's simplify each term:
y = x^4/16 + 11x^2/16 + 11x^2/16 + 121/16
Hmm, we've got a couple of similar terms to combine:
y = x^4/16 + 22x^2/16 + 121/16
And there you have it! The simplified expression for y = f(x) is:
y = (x^4 + 22x^2 + 121)/16
So, my friend, your initial expression was a bit of a mouthful, but we simplified it down to a neat and tidy expression. Keep up the good work, and remember, math can be fun, even for a clown like me!
To find the expression for y = f(x) by solving the differential equation dy/dx = x√y with the initial condition f(3) = 25, follow these steps:
Step 1: Separate the variables.
dy/√y = xdx
Step 2: Integrate both sides.
∫(1/√y)dy = ∫xdx
The integral of (1/√y)dy can be evaluated as 2√y, and the integral of xdx is (1/2)x^2.
So we have:
2√y = (1/2)x^2 + C, where C is the constant of integration.
Step 3: Solve for y.
To isolate y in terms of x, square both sides of the equation:
(2√y)^2 = [(1/2)x^2 + C]^2
4y = (1/4)x^4 + C^2 + x^2C + C(1/2)x^2
Simplifying further, we get:
4y = (1/4)x^4 + C^2 + (3/2)C(x^2)
Step 4: Apply the initial condition.
Given f(3) = 25, substitute x = 3 and y = 25 into our expression for y:
4(25) = (1/4)(3^4) + C^2 + (3/2)C(3^2)
100 = (81/4) + C^2 + 27C
Rearranging and simplifying the equation, we solve for C:
100 - (81/4) - 27C = C^2
C^2 + 27C - (400 - 81/4) = 0
C^2 + 27C - (1619/4) = 0
Step 5: Solve for C.
Use the quadratic formula to solve for C:
C = (-27 ± √(27^2 - 4(1)(-1619/4))) / (2(1))
C = (-27 ± √(729 + 6476/4)) / 2
C = (-27 ± √(729 + 1619)) / 2
C = (-27 ± √2348) / 2
C = (-27 ± √4(587)) / 2
C = (-27 ± 2√587) / 2
C = -13.5 ± √587
Since C cannot be negative, we have:
C ≈ 16.2
Step 6: Substitute C back into the expression for y.
Plugging C = 16.2 back into the equation we found in Step 4:
4y = (1/4)x^4 + (16.2)^2 + (3/2)(16.2)(x^2)
y = [(1/16)x^4 + 16.2x^2 + 16.2^2]
Simplifying further, we get:
y = (x^4/16 + 27.24x^2 + 262.44)
Therefore, the expression for y = f(x) by solving the given differential equation with the initial condition f(3) = 25 is:
y = (x^4/16 + 27.24x^2 + 262.44)
To solve the given differential equation, we will separate the variables and integrate both sides.
Start with the given differential equation:
dy/dx = x√y
Separate the variables by multiplying both sides by dx and dividing both sides by √y:
√y dy = x dx
Now, integrate both sides with respect to their respective variables:
∫√y dy = ∫x dx
To integrate the left side, use the power rule for integration. Substitute u = √y and du = (1/2) y^(-1/2) dy:
∫√y dy = ∫2u du = u^2 + C1
To integrate the right side, use the power rule for integration:
∫x dx = (1/2) x^2 + C2
Combining both sides and removing the constant of integration for simplicity, we have:
u^2 = (1/2) x^2
Substitute back u = √y:
(√y)^2 = (1/2) x^2
Simplifying:
y = (1/2) x^2
Now, apply the initial condition f(3) = 25 to find the specific solution. Plug in x = 3 and y = 25:
25 = (1/2) (3)^2
25 = (1/2) (9)
25 = 4.5
Since the initial condition does not hold, the expression y = (x^2/4 + 11/4)^2 is incorrect.
To find the correct expression for y = f(x), follow the steps above and solve for y again.