Differential Equations Spring 2023 - Homework 11 Feel free to use a calculator in this assignment, but be sure to show all of your work. Topics: Applications of DEs, Numerical Solutions Due: Tuesday May 2nd, 11:59 pm (on Gradescope). 1. Make the following approximations using Euler’s Method. (a) y ′ = xy + 1 , y (0) = 1 , h = 0 25 , y (1) =? (b) y ′ = sin( x ) + cos( y ) , y (0) = 0 , h = π/ 4 , y ( π ) =? 2. Reduce the order of the given DE using u = y ′ and use Euler’s Method to make the given approxima- tion. (a) y ′′ = x + y + y ′ , y (0) = 1 , y ′ (0) = 1 , h = 0 5 , y (1 5) =? (b) y ′′ = e y + xy ′ , y (0) = 0 , y ′ (0) = 0 , h = 0 5 , y (1 5) =? 3. (a) Improved Euler’s Method uses not only the derivative at x n but the estimated derivative at x n +1 . It has the following formula for y n +1 (the one for x n +1 is the same as Euler’s Method): y n +1 = y n + ( h/ 2)( f ( x n , y n ) + f ( x n +1 , y n + k )) , where y ′ = f ( x, y ) and k is the value for y n +1 using Euler’s Method. Approximate y (1) given that y ′ = y, y (0) = 1 , h = 0 5. (b) Use an online Euler’s method calculator to see how small h would have to be to get the same level of accuracy as your calculation in part a using regular Euler’s Method. 4. Tank T 1 initially contain 50 gallons of pure water. Starting at t = 0 , water that contains 1 pound of salt per gallon is poured into T 1 at the rate of 2 gal/min. The mixture is drained from T 1 at the same rate into a second tank T 2 , which initially contains 50 gallons of pure water. Also starting at t = 0, a mixture from another source that contains 2 pounds of salt per gallon is poured into T 2 at the rate of 2 gal/min. The mixture is drained from T 2 at the rate of 4 gal/min. Find and solve a differential equation for Q ( t ), the amount of salt in T 2 . What happens to Q ( t ) as t goes to infinity? 5. Recall that for a RLC circuit, the charge as a function of time can be given by the DE LQ ′′ + RQ ′ + Q/C = E ( t ), where L is the inductance, R is the resistance, C is the capacitance, and E is the impressed voltage. Suppose L = 1 , R = 4 , C = 1 / 4, and E ( t ) = u ( t ) − u ( t − 1). If Q (0) = 1 and Q ′ (0) = 0, find a function for the current, I ( t ) = Q ′ ( t ). 1