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

 Solve the following differential equation, 𝑑𝑥𝑑𝑡 =9𝑥𝑒11𝑥+𝑡. 

Let x(t) be the solution with initial value 𝑥(0)=1. If 𝑡0 = (−ln(111089)𝑒11+1), find the value of 𝑥(𝑡0) with the answer rounded to exactly 4 digits after the decimal point. 

[1] Always identify the type of ODE given prior to solving. This will not only maximize points on exams, it will give you the formula for the solution. 

This is a first order separable differential equation. (It is a Calculus II question and nothing more.) 

𝑓(𝑥,𝑡)=𝑑𝑥𝑑𝑡 ⇔ 𝑓(𝑥)𝑔(𝑡)=𝑑𝑥𝑑𝑡 ⇔ ∫𝑔(𝑡)𝑑𝑡=∫𝑓(𝑥)𝑑𝑥 

See how it breaks into two basic integrals. 

NOTE: 𝑑𝑥𝑑𝑡 means 𝑥 is the dependent variable and 𝑡 is the independent. We solve for 𝑥(𝑡). 

[2] Apply the formula. 

𝑑𝑥𝑑𝑡 =9𝑥𝑒11𝑥+𝑡 ⇒ 𝑑𝑥𝑑𝑡 =9𝑥𝑒11𝑥𝑒𝑡 ⇒ 𝑥𝑒11𝑥𝑑𝑥=9𝑒−𝑡𝑑𝑡 

⇒ ∫𝑥𝑒11𝑥𝑑𝑥=∫9𝑒−𝑡𝑑𝑡 

You will do IBPs on the left. 

111𝑒11 𝑥 𝑥 −𝑒11 𝑥121+𝑐1=−9𝑒−𝑡+𝑐2 ⇒ 1121𝑒11 𝑥( 11𝑥−1)=−9𝑒−𝑡+𝐶 

[3] Solve for 𝐶 by evaluating the solution at 𝑥(0)=1. 

1121𝑒11 (1)( 11(1)−1)=−9𝑒−(0)+𝐶 ⇒ 𝐶 = 9 +10 𝑒11121 

Now, just evaluate 1121𝑒11 𝑥( 11𝑥−1)=−9𝑒−𝑡+9 +10 𝑒11121 at 𝑡0 = (−ln(111089)𝑒11+1) you will need to use whatever your professor wants for calculator. Could be MATLAB or something related… I will leave this part to you.