find the general solution of the following first order differential equation.
dx/dt = t*x + 6*t(exp)-t^2
pls show working
To find the general solution of the given first-order differential equation, we will use the method of integrating factors.
Given the equation:
dx/dt = t*x + 6*t(exp)(-t^2)
Step 1: Rearrange the equation into standard form
dx/dt - t*x = 6*t(exp)(-t^2)
Step 2: Identify the integrating factor, which is denoted by μ(t). The integrating factor is calculated by taking the exponential of the integral of the coefficient of x. In this case, the coefficient of x is -t.
μ(t) = exp ∫(-t) dt
= exp (-t^2/2)
Step 3: Multiply both sides of the equation by the integrating factor μ(t)
(exp (-t^2/2))*dx/dt - t*(exp (-t^2/2))*x = 6*t*(exp (-t^2/2))
Step 4: Apply the product rule to simplify the left side of the equation. The product rule states that d(uv)/dt = u*(dv/dt) + v*(du/dt), where u and v are functions of t.
d(μ(t)*x)/dt = d(μ(t))/dt * x + μ(t) * dx/dt
Differentiating μ(t) with respect to t using the power rule:
d(μ(t))/dt = exp (-t^2/2) * (-t)
Substituting the values into the equation:
exp (-t^2/2) * dx/dt - t * exp (-t^2/2) * x = 6 * t * exp (-t^2/2)
Step 5: Rewrite the equation in the simplified form:
d(μ(t)*x)/dt = -t * exp (-t^2/2)
Step 6: Integrate both sides of the equation with respect to t.
∫ d(μ(t)*x)/dt dt = ∫ (-t) * exp (-t^2/2) dt
Integrating the left side gives us:
μ(t) * x = - ∫ t * exp (-t^2/2) dt
Step 7: Solve the integral.
To solve the integral, we will use the substitution method. Let u = -t^2/2, then du/dt = -t.
- ∫ t * exp (-t^2/2) dt = ∫ exp(u) du
Integrating exp(u) with respect to u gives:
- ∫ exp(u) du = - exp(u)
Substituting the value back:
μ(t) * x = -(- exp(-t^2/2)) + C
Simplifying:
μ(t) * x = exp(-t^2/2) + C
Step 8: Divide both sides of the equation by μ(t) to solve for x.
x = (exp(-t^2/2) + C) / μ(t)
Therefore, the general solution to the given first-order differential equation is:
x(t) = (exp(-t^2/2) + C) / exp (-t^2/2)
or
x(t) = exp(t^2/2)(exp(-t^2/2) + C)
where C is an arbitrary constant.