Final answer:
To solve the differential equation dx/dt = x ln(x)/t, we can separate the variables and integrate both sides. After integrating, we can use the initial condition to find the constants of integration and obtain the particular solution of the equation.
Step-by-step explanation:
The given differential equation is dx/dt = x ln(x)/t. To solve this equation, we can separate the variables and integrate both sides. Let's start by moving the t term to the other side:
dt = (x ln(x))/x
Integrating both sides gives:
∫ dt = ∫ (ln(x))/x dx
Integrating the left side gives t + C₁, where C₁ is the constant of integration. For the right side, we use integration by parts:
u = ln(x), dv = dx/x
du = dx/x, v = ln(x)
Using the formula for integration by parts: ∫ u dv = uv - ∫ v du
we have:
ln(x) ln(x) - ∫ ln(x) * (dx/x)
Integrating ∫ ln(x) * (dx/x) gives us ∫ ln(x) dx = x(ln(x) - 1) + C₂
Substituting back into the equation gives:
t + C₁ = x(ln(x) - 1) + C₂
Given the initial condition x(1) = 5, we can substitute t = 1 and x = 5 into the equation to find C₁ and C₂:
1 + C₁ = 5(ln(5) - 1) + C₂
From here, we can simplify and solve for C₁ and C₂ to find the particular solution of the differential equation.