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.