Given:
dy/dx = 2x^3 √y - 4xy
y(0) = 0

To find:
y(1)

Explanation:

To solve the given differential equation, we can use the method of separation of variables. Let's break down the solution into steps:

Step 1: Rewrite the equation

dy/dx = 2x^3 √y - 4xy

Step 2: Separate variables

Divide both sides of the equation by (√y) to isolate y terms on one side and x terms on the other side:

1/√y dy = 2x^3 - 4x√y dx

Step 3: Integrate both sides

Integrate both sides of the equation with respect to x:

∫(1/√y) dy = ∫(2x^3 - 4x√y) dx

Simplifying the integrals:

2∫x^3 dx - 4∫x√y dx = ∫(1/√y) dy

Integrating each term:

(1/2)x^4 - 4(2/3)x^(3/2)√y = 2√y + C

Where C is the constant of integration.

Step 4: Apply the initial condition

Using the initial condition y(0) = 0, we can find the value of C:

(1/2)(0)^4 - 4(2/3)(0)^(3/2)√0 = 2√0 + C

Simplifying:

0 - 0 = 0 + C

C = 0

Step 5: Substitute the value of C

Substitute the value of C back into the equation:

(1/2)x^4 - 4(2/3)x^(3/2)√y = 2√y + 0

Simplifying further:

(1/2)x^4 - 4(2/3)x^(3/2)√y = 2√y

Step 6: Solve for y(1)

To find y(1), substitute x = 1 into the equation:

(1/2)(1)^4 - 4(2/3)(1)^(3/2)√y = 2√y

Simplifying:

1/2 - 8/3√y = 2√y

Rearranging the terms:

8/3√y + 2√y = 1/2

Combining like terms:

14/3√y = 1/2

Squaring both sides:

(14/3√y)^2 = (1/2)^2

Simplifying:

196/9y = 1/4

Cross-multiplying:

4 * 196 = 9y

784 = 9y

Dividing by 9:

y = 784/9

Therefore, y(1) = 784/9.