Question

If \( x(t) \) is the solution to the differential equation \[ \frac{dx}{dt} = x t^3 + x t, \text{ for } t > 0, \text{ satisfying } x(0) = 1, \] then the value of \( x(\sqrt{2}) \) is .......... (correct up to two decimal places). 
 

Hint:

For separable differential equations, separate variables, integrate both sides, and solve using initial conditions.

Answer 1

Correct Answer: -2.8

Step 1: Rewrite the differential equation.
We are given the differential equation \( \frac{dx}{dt} = x t^3 + x t \). This can be written as: \[ \frac{dx}{dt} = x(t) (t^3 + t). \] This is a separable differential equation.

Step 2: Separate variables.
We separate the variables and integrate: \[ \frac{1}{x(t)} \, dx = (t^3 + t) \, dt. \]

Step 3: Integrate both sides.
Integrating both sides: \[ \int \frac{1}{x(t)} \, dx = \int (t^3 + t) \, dt, \] \[ \ln |x(t)| = \frac{t^4}{4} + \frac{t^2}{2} + C. \]

Step 4: Solve for the constant.
Using the initial condition \( x(0) = 1 \), we solve for \( C \): \[ \ln 1 = 0 + 0 + C $\Rightarrow$ C = 0. \]

Step 5: Solve for \( x(t) \).
Thus, we have: \[ x(t) = e^{\frac{t^4}{4} + \frac{t^2}{2}}. \]

Step 6: Calculate \( x(\sqrt{2}) \).
Substituting \( t = \sqrt{2} \) into the expression for \( x(t) \), we get: \[ x(\sqrt{2}) = e^{\frac{(\sqrt{2})^4}{4} + \frac{(\sqrt{2})^2}{2}} = e^{\frac{4}{4} + \frac{2}{2}} = e^{1 + 1} = e^2. \] Thus, \( x(\sqrt{2}) \approx \boxed{7.389} \).