Solve the given differential equation.

Question

dx/dt =(9/(xe^(11x+t))

Let x(t) be the solution with initial value x(0) = 1. If t0 = -ln(10/1089)e^11+1), find the value of x (t0) with the answer rounded to exactly 4 digits after the decimal point.

Answer 1

To solve this initial value problem with the given differential equation:

dx/dt = (9 / (x * e^(11x + t)))

and the initial condition x(0) = 1, we can separate the variables and solve it.

First, separate variables:

dx / (x * e^(11x + t)) = 9 dt

Now, integrate both sides:

∫ (1 / (x * e^(11x + t))) dx = ∫ 9 dt

Let's solve the left integral first. We can use a substitution here. Let u = 11x + t, which means du = 11 dx.

So, the integral becomes:

(1 / 11) ∫ (1 / (x * e^u)) du

Now, we have:

(1 / 11) ∫ (e^(-u) / x) du

Integrating this with respect to u:

(1 / 11) * (-ln|x| - u) + C₁

Now, let's solve the right integral:

∫ 9 dt = 9t + C₂

Now, we can combine the two parts:

(1 / 11) * (-ln|x| - u) + C₁ = 9t + C₂

Now, we apply the initial condition x(0) = 1, which gives us:

(1 / 11) * (-ln|1| - 0) + C₁ = 9 * 0 + C₂

(1 / 11) * (0) + C₁ = 0 + C₂

C₁ = C₂

So, our equation becomes:

(1 / 11) * (-ln|x| - u) + C = 9t + C

Now, we can use the value of t₀ = -ln(10/1089)e^11+1) to find x(t₀).

(1 / 11) * (-ln|x| - u) + C = 9 * (-ln(10/1089)e^11+1))

Now, let's solve for x at t₀:

-ln|x| - u = 99 * (-ln(10/1089)e^11+1))

-ln|x| - (11x + t₀) = 99 * (-ln(10/1089)e^11+1))

Now, plug in the value of t₀:

-ln|x| - (11x - ln(10/1089)e^11+1)) = 99 * (-ln(10/1089)e^11+1))

Now, let's solve for |x|:

ln|x| = 11x - ln(10/1089)e^11+1) - 99 * (-ln(10/1089)e^11+1))

ln|x| = 11x + ln(10/1089)e^11+1) + 99 * ln(10/1089)e^11+1)

Now, take the exponential of both sides:

|x| = e^(11x + ln(10/1089)e^11+1) + 99 * ln(10/1089)e^11+1)

|x| = e^(11x) * (10/1089)e^11+1)^(99)

Now, we can find the value of x at t₀:

x(t₀) = e^(11 * (-ln(10/1089)e^11+1))) * (10/1089)e^11+1)^(99)

Calculate this expression, and you'll find the value of x(t₀). Be sure to round it to exactly 4 digits after the decimal point.