A differentiable function y = g(x) satisfies `int_0^x(x-t+1) g(t) dt=x^4+x^2` for all `x>=0` then y=g(x) satisfies the differential equation

Options

A. `(dy)/(dx) -y = 12 x^(2) +2`
B. `(dy)/(dx)+2y =12 x ^(2) +2`
C. `(dy)/(dx )+ y =12 x ^(2) +2`
D. `(dy)/(dx) +y =12x +2`

Answer

To solve the problem, we need to find the differential equation satisfied by the function \( y = g(x) \) given the integral condition:

\[ \int_0^x (x - t + 1) g(t) \, dt = x^4 + x^2 \]

for all \( x \geq 0 \).

### Step 1: Simplify the Integral Expression

Starting with the given integral, we can rewrite it as:

\[ \int_0^x (x - t + 1) g(t) \, dt = \int_0^x (x g(t) - t g(t) + g(t)) \, dt \]

This can be split into three separate integrals:

\[ = \int_0^x x g(t) \, dt - \int_0^x t g(t) \, dt + \int_0^x g(t) \, dt \]

### Step 2: Factor Out Constants

Since \( x \) is treated as a constant when integrating with respect to \( t \), we can factor \( x \) out of the first integral:

\[ = x \int_0^x g(t) \, dt - \int_0^x t g(t) \, dt + \int_0^x g(t) \, dt \]

### Step 3: Combine the Integrals

Now we can combine the terms:

\[ = x \int_0^x g(t) \, dt - \int_0^x t g(t) \, dt + \int_0^x g(t) \, dt = x \int_0^x g(t) \, dt + \int_0^x g(t) \, dt - \int_0^x t g(t) \, dt \]

### Step 4: Set the Equation Equal to the Right Side

Now, we equate this to the right-hand side of the original equation:

\[ x \int_0^x g(t) \, dt + \int_0^x g(t) \, dt - \int_0^x t g(t) \, dt = x^4 + x^2 \]

### Step 5: Differentiate Both Sides

Next, we differentiate both sides with respect to \( x \):

Using Leibniz's rule for differentiation under the integral sign, we get:

\[ \frac{d}{dx} \left( x \int_0^x g(t) \, dt \right) + \frac{d}{dx} \left( \int_0^x g(t) \, dt \right) - \frac{d}{dx} \left( \int_0^x t g(t) \, dt \right) = \frac{d}{dx}(x^4 + x^2) \]

Calculating the derivatives:

1. For the first term:
\[ \int_0^x g(t) \, dt + x g(x) \]
2. For the second term:
\[ g(x) \]
3. For the third term:
\[ x g(x) - \int_0^x t g'(t) \, dt \]
4. For the right side:
\[ 4x^3 + 2x \]

Combining these gives:

\[ \int_0^x g(t) \, dt + x g(x) + g(x) - (x g(x) - \int_0^x t g'(t) \, dt) = 4x^3 + 2x \]

### Step 6: Simplify the Equation

After simplification, we find:

\[ \int_0^x g(t) \, dt + g(x) + \int_0^x t g'(t) \, dt = 4x^3 + 2x \]

### Step 7: Differentiate Again

Differentiating again gives us:

\[ g(x) + g'(x) = 12x^2 + 2 \]

### Step 8: Rearranging to Form the Differential Equation

This can be rearranged to form the differential equation:

\[ \frac{dy}{dx} + y = 12x^2 + 2 \]

### Final Result

Thus, the differential equation satisfied by \( y = g(x) \) is:

\[ \frac{dy}{dx} + y = 12x^2 + 2 \]