Question

Solve the given initial value problem xy² dy/dx=y³-x³
y(1)=4

Answer 1

After solving the differential equation using the given initial condition (y(1) = 4\), the particular solution is:

[y = {4x^4}/{4x^4+1}\]

Step-by-step explanation:

The question asks us to solve the initial value problem (IVP) with the given differential equation xy² dy/dx=y³-x³ and the initial condition y(1)=4. This is a first-order non-linear ordinary differential equation (ODE). To solve this, we will need to apply separation of variables and initial conditions to find the solution that passes through the given point (1, 4).

First, we rewrite the ODE in a separable form:

1. Rearrange the equation to isolate the differential terms: xy² dy = (y³ - x³) dx.
2. Divide both sides by y³ and multiply by dx to get (1/y) dy = (1/x - x²/y³) dx.
3. Integrate both sides to find the general solution of the differential equation.
4. Apply the initial condition y(1) = 4 to find the particular solution.

The resulting explicit solution will depend on the integration and application of the initial condition.

Answer 2

The solution to the given initial value problem is y(x) =
`(x^2 + 4)^(1/2)`.

Step-by-step explanation:

To solve the given initial value problem, we start by rearranging the differential equation:

`\[ xy^2 (dy)/(dx) = y^3 - x^3 \]`

Divide both sides by
`\(xy^2\)` to separate variables:

`\[ (1)/(y^2) (dy)/(dx) = (y)/(x) - (x^2)/(y^2) \]`

Now, let's make the substitution
`(v = (1)/(y)\)`. Then,
`\((dv)/(dx) = -(1)/(y^2) (dy)/(dx)\)` . Substitute this into the equation:

`\[ (dv)/(dx) = -(v)/(x) + x^2v^2 \]`

This is a first-order linear differential equation, and we can solve it using an integrating factor. The solution is:

`\[ v(x) = (1)/((x^2 + C)^(1/2)) \]`

Now, substitute back
`\(y = (1)/(v)\)`:

`\[ y(x) = (x^2 + C)^(1/2) \]`

Apply the initial condition (y(1) = 4) to solve for (C), yielding the final solution:

`\[ y(x) = (x^2 + 4)^(1/2) \]`