ENEL101 Problem set 7 Symbolic Math

Description

Q1. Using Matlab symbolic math and the solve() function, find the leftmost point (with x on the
horizontal axis) where these two ellipses intersect:
(𝑥 − 1 + 𝑟)(
6( + 𝑦(
3( = 1
𝑥 + 2 (
2( + 𝑦 − 5 (
4( = 1
where 𝑟 = 0.1. Please make your answer a 1×2 vector with the x and y components of the
answer. Hint 1) Due to the presence of variable 𝑟, first create symbolic expressions for the
ellipses and then use them in solve(), rather than putting it in quotes as an argument to solve().
Hint 2) Use solve(ellipse1,ellipse2,’MaxDegree’,4) to make MATLAB give you an analytical
solution Hint 3) use eval() to turn your symbolic answers into floating-point answers
ENEL101 Assignment 6 Page 2
Q2. Write a function that finds the equation of the tangent line to the upper part of the ellipse (x
on horizontal axis, y on vertical axis)
𝑥 + 2 (
2( + 𝑦 − 5 (
4( = 1
at 𝑥4 = −0.1. The answer will be a symbolic math expression of the form
𝑚𝑥 + 𝑏
where 𝑚 and 𝑏 have numerical values and 𝑥 is a symbolic variable. Hint: first solve for 𝑦 using
solve(), select the 𝑦 that is the top part of the ellipse, then use diff() to take the derivative with
respect to 𝑥, then get the slope 𝑚 by substituting in 𝑥4 using subs(), and then calculate the yintercept 𝑏. As the very last step you can use vpa() to turn all numerical values in the symbolic
answer (i.e. 𝑚 and 𝑏) into floating point numbers in order to compare your answer to the solution
set.
Q3. Evaluate the following indefinite integral
𝐼 = 𝑥8
𝑅 − 𝑥(
𝑑𝑥
for R=1. Hint: use the function subs() to substitute a numerical value for R.
Q4. The current 𝑖 in a series RLC circuit can be described the differential equation
𝐿
𝑑(𝑖
𝑑 𝑡( + 𝑅
𝑑𝑖
𝑑𝑡 +
1
𝐶
𝑖 = 0
Using dsolve(), find the solution 𝑖 𝑡 given initial conditions 𝑖 0 = 0, ?@
?A (0) = 8, and parameters 𝐿 =
3𝐻, 𝑅 = 10Ω, 𝐶 = 80𝜇𝐹. Use variable ‘current’ in the code instead of ‘i’ to avoid conflicts with ‘i’ as a
numerical counter.