University:
University of California, Berkeley
Course:
MATH 51 | Calculus I
Academic year:

2023
Views:

24

Pages:

1
Author:

Clint W.

Section 4.7 Definition: A linear differential equation of the form n n−1 d y dy n−1 d an xn dx n + an−1 x dxn−1 + · · · + a1 x dx + a0 y = g(x), where the coefficients an , an−1 ,...., a0 are constants, is known as a Cauchy-Euler equation. Fact: y = xm is a solution of the differential equation whenever m is a solution of the auxiliary equation am(m − 1) + bm + c = 0. Case 1: Distinct Real Roots: Let m1 and m2 denote the real roots of the auxiliary equation. The general solution is y = c1 xm1 + c2 xm2 . Case 2: Repeated Real Roots: If the roots of the auxiliary equation are repeated and m1 is the root, then the general solution is y = c1 xm1 + c2 xm1 ln x. Case 3: Conjugate Complex Roots: If roots of the auxiliary equation are m1 = α + iβ and m2 = α − iβ, then the general solution is y = xα [c1 cos(β ln x) + c2 sin (β ln x) ]. Exercise 1. Find two linearly independent solutions to the equation 3t2 y ′′ + 11ty ′ − 3y = 0, t > 0. (Nagle Section 4.7 Example 2) Exercise 2. Find a pair of linearly independent solutions to the following Cauchy-Euler equation for t > 0. (a) t2 y ′′ + 5ty ′ + 5y = 0 (b) t2 y ′′ + ty ′ = 0 (Nagle Section 4.7 Example 3) Exercise 3. Solve the differential equation 4x2 y ′′ + y = 0. (#2) Class Exercise 1. Solve the differential equation xy ′′ − 3y ′ = 0. (#4) Exercise 4. Solve the differential equation x2 y ′′ + 5xy ′ + 3y = 0. (#6) Class Exercise 2. Solve the given differential equation. (#8, 10, 12, 14, 16) (a) x2 y ′′ + 3xy ′ − 4y = 0 (b) 4x2 y ′′ + 4xy ′ − y = 0 (c) x2 y ′′ + 8xy ′ + 6y = 0 2 ′′ ′ 3 ′′′ ′ (d) x y − 7xy + 41y = 0 (e) x y + xy − y = 0 Exercise 5. Solve the differential equation 2x2 y ′′ + 5xy ′ + y = x2 − x by variation of parameters. (#20) Class Exercise 3. Solve the differential equation by variation of parameters. (#22, 24) 1 (a) x2 y ′′ − 2xy ′ + 2y = x4 ex (b) x2 y ′′ + xy ′ − y = x+1 Exercise 6. Use the substitution x = et to transform the Cauchy-Euler equation x2 y ′′ − 9xy ′ + 25y = 0 to a differential equation with constant coefficients. Then, solve the equation. (#32) Note: If x = et , dy dx = 1 dy x dt and d2 y dx2 = 1 x2 2 ( ddt2y − dy dt ) Class Exercise 4. Use the substitution x = et to transform the Cauchy-Euler equation to a differential equation with constant coefficients. Then, solve the equation. (#34, 36) (a) x2 y ′′ − 4xy ′ + 6y = ln x2 (b) x3 y ′′′ − 3x2 y ′′ + 6xy ′ − 6y = 3 + ln x3