Question

If \((t+1)dx=(2x+(t+1)^3)dt\) and \(x(0)=2\), then \(x(1) \)is equal to:

• A. 5
• B. 6
• C. 12
• D. 8

Answer 1

The Correct Option is C

We are given the differential equation:

((t+1)dx = (2x+(t+1)^3)dt)

with the initial condition x(0) = 2. We are asked to find x(1).

To solve this, we will separate the variables and integrate.

Step 1: Separate the variables.

Rewriting the given equation:

(t+1)dx = (2x + (t+1)^3)dt

We can rearrange it to:

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

Now, separate the variables x and t:

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

Step 2: Integrate the separated variables.

We rewrite the equation to facilitate integration:

\frac{dx}{x} = \frac{2}{t+1} dt + (t+1)^2 dt

Now we integrate each side:

\int \frac{dx}{x} = \int \frac{2}{t+1} dt + \int (t+1)^2 dt

This gives us:

\ln|x| = 2\ln|t+1| + \frac{(t+1)^3}{3} + C

Convert the equation to exponential form:

x = e^{C} \cdot (t+1)^2 \cdot e^{\frac{(t+1)^3}{3}}

Step 3: Apply the initial condition.

Using the initial condition x(0) = 2, we find C:

x(0) = e^{C} \cdot 1^2 \cdot e^{\frac{1^3}{3}} = 2

Thus:

e^{C + \frac{1}{3}} = 2

e^{C} = 2 \cdot e^{-\frac{1}{3}}

Step 4: Calculate x(1).

Now substitute t = 1 into the equation:

x(1) = e^{C} \cdot (1+1)^2 \cdot e^{\frac{(1+1)^3}{3}}

x(1) = 2 \cdot e^{-\frac{1}{3}} \cdot 4 \cdot e^{\frac{8}{3}}

x(1) = 2 \cdot 4 \cdot e^{\frac{8-1}{3}}

x(1) = 8 \cdot e^{3/3}

x(1) = 8 \cdot e^1

The value of x(1) is 12.