If `x = x(t)` is the solution of the differential equation `(t + 1)dx = (2x + (t + 1)^4) dt, x(0) = 2`, then, x(1) equals ___________.

Answer

To solve the given differential equation and find \( x(1) \), we start with the equation:

\[ (t + 1)dx = (2x + (t + 1)^4)dt \]

### Step 1: Rewrite the equation
We can rewrite the equation in the standard form of a first-order linear differential equation by dividing both sides by \( t + 1 \):

\[ dx = \left( \frac{2x}{t + 1} + (t + 1)^3 \right) dt \]

### Step 2: Isolate \( dx/dt \)
Now, we express this as:

\[ \frac{dx}{dt} - \frac{2x}{t + 1} = (t + 1)^3 \]

This is now in the form of \( \frac{dx}{dt} + P(t)x = Q(t) \) where \( P(t) = -\frac{2}{t + 1} \) and \( Q(t) = (t + 1)^3 \).

### Step 3: Find the integrating factor
The integrating factor \( \mu(t) \) is given by:

\[ \mu(t) = e^{\int P(t) dt} = e^{\int -\frac{2}{t + 1} dt} = e^{-2 \ln(t + 1)} = (t + 1)^{-2} \]

### Step 4: Multiply through by the integrating factor
Multiplying the entire differential equation by the integrating factor:

\[ (t + 1)^{-2} \frac{dx}{dt} - \frac{2x}{(t + 1)^3} = (t + 1)^{-2} (t + 1)^3 \]

This simplifies to:

\[ (t + 1)^{-2} \frac{dx}{dt} - \frac{2x}{(t + 1)^3} = (t + 1) \]

### Step 5: Rewrite the left side
The left side can be rewritten as:

\[ \frac{d}{dt} \left( x (t + 1)^{-2} \right) = (t + 1) \]

### Step 6: Integrate both sides
Integrating both sides with respect to \( t \):

\[ \int \frac{d}{dt} \left( x (t + 1)^{-2} \right) dt = \int (t + 1) dt \]

This gives:

\[ x (t + 1)^{-2} = \frac{t^2}{2} + t + C \]

### Step 7: Solve for \( x \)
Multiplying through by \( (t + 1)^2 \):

\[ x = \left( \frac{t^2}{2} + t + C \right)(t + 1)^2 \]

### Step 8: Apply the initial condition
Using the initial condition \( x(0) = 2 \):

\[ 2 = \left( \frac{0^2}{2} + 0 + C \right)(0 + 1)^2 \]

This simplifies to:

\[ 2 = C \implies C = 2 \]

### Step 9: Substitute \( C \) back into \( x \)
Now substituting \( C \) back into the equation:

\[ x = \left( \frac{t^2}{2} + t + 2 \right)(t + 1)^2 \]

### Step 10: Find \( x(1) \)
Now we need to find \( x(1) \):

\[ x(1) = \left( \frac{1^2}{2} + 1 + 2 \right)(1 + 1)^2 \]

Calculating this gives:

\[ x(1) = \left( \frac{1}{2} + 1 + 2 \right)(2^2) = \left( \frac{1}{2} + 3 \right)(4) = \left( \frac{1}{2} + \frac{6}{2} \right)(4) = \left( \frac{7}{2} \right)(4) = 14 \]

Thus, the final answer is:

\[ \boxed{14} \]