Questions: (d) dx/dt = 2tx + 4e^(-t^2), x(0) = 10.

Solution Steps

To solve the given differential equation, we can use numerical methods such as the Runge-Kutta method. The equation is a first-order ordinary differential equation (ODE) with an initial condition. We will use the scipy.integrate.solve_ivp function in Python to solve this ODE.

The given differential equation is: \[ \frac{d x}{d t} = 2 t x + 4 e^{-t^2} \] with the initial condition \( x(0) = 10 \).

This is a first-order linear differential equation of the form: \[ \frac{d x}{d t} + P(t)x = Q(t) \] where \( P(t) = -2t \) and \( Q(t) = 4e^{-t^2} \).

The integrating factor \( \mu(t) \) is given by: \[ \mu(t) = e^{\int P(t) \, dt} = e^{\int -2t \, dt} = e^{-t^2} \]

Multiplying both sides of the differential equation by \( e^{-t^2} \): \[ e^{-t^2} \frac{d x}{d t} + e^{-t^2} (2 t x) = 4 e^{-t^2} e^{-t^2} \] \[ e^{-t^2} \frac{d x}{d t} + 2 t x e^{-t^2} = 4 e^{-2t^2} \]

Notice that the left-hand side is the derivative of \( x e^{-t^2} \): \[ \frac{d}{dt} \left( x e^{-t^2} \right) = 4 e^{-2t^2} \]

Integrate both sides with respect to \( t \): \[ \int \frac{d}{dt} \left( x e^{-t^2} \right) \, dt = \int 4 e^{-2t^2} \, dt \] \[ x e^{-t^2} = \int 4 e^{-2t^2} \, dt \]

The integral on the right-hand side is not elementary, but we can express it in terms of the error function \( \text{erf}(t) \): \[ \int 4 e^{-2t^2} \, dt = 2 \sqrt{\pi} \, \text{erf}(\sqrt{2} t) + C \]

Using the initial condition \( x(0) = 10 \): \[ x(0) e^{-0^2} = 2 \sqrt{\pi} \, \text{erf}(0) + C \] \[ 10 = 0 + C \] \[ C = 10 \]

Substitute \( C \) back into the equation: \[ x e^{-t^2} = 2 \sqrt{\pi} \, \text{erf}(\sqrt{2} t) + 10 \] \[ x = e^{t^2} \left( 2 \sqrt{\pi} \, \text{erf}(\sqrt{2} t) + 10 \right) \]

Final Answer