In this problem you will use undetermined coefficients to solve the nonhomogeneous equation y" + 4y + 4y = 6e-2-12te-2 +8t+20 with initial values y(0) = 2 and y'(0) = 3. A. Write the characteristic equation for the associated homogeneous equation. (Use for your variable.) (r+2)²=0 B. Write the fundamental solutions for the associated homogeneous equation. 3/1 C. Write the form of the particular solution and its derivatives. (Use ABC etc. for undetermined coefficients. Y2 = te Y At³e-2t+Bet+Ct+D -21 Y' = A(-21³ +31²) e 2t+B(-21² +2t) e 2t+C Y" = 4(4³-12² +6t)e-2t + B(4t² - 8t+2) ₁-²t D. Write the general solution. (Use c1 and c2 for c₁ and c₂). y= E. Plug in the initial values and solve for c₁ and c₂ to find the solution to the initial value problem. y= Hint: No fractions are required in the solution or answer to this problem
In this problem you will use undetermined coefficients to solve the nonhomogeneous equation y" + 4y + 4y = 6e-2-12te-2 +8t+20 with initial values y(0) = 2 and y'(0) = 3. A. Write the characteristic equation for the associated homogeneous equation. (Use for your variable.) (r+2)²=0 B. Write the fundamental solutions for the associated homogeneous equation. 3/1 C. Write the form of the particular solution and its derivatives. (Use ABC etc. for undetermined coefficients. Y2 = te Y At³e-2t+Bet+Ct+D -21 Y' = A(-21³ +31²) e 2t+B(-21² +2t) e 2t+C Y" = 4(4³-12² +6t)e-2t + B(4t² - 8t+2) ₁-²t D. Write the general solution. (Use c1 and c2 for c₁ and c₂). y= E. Plug in the initial values and solve for c₁ and c₂ to find the solution to the initial value problem. y= Hint: No fractions are required in the solution or answer to this problem
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Just answer E and D
Transcribed Image Text:In this problem you will use undetermined coefficients to solve the nonhomogeneous equation
y" + 4y + 4y = 6e-²-12te-2 +8t+20
with initial values y(0) = 2 and y(0) = 3.
A. Write the characteristic equation for the associated homogeneous equation. (Use for your variable.)
|(x+2)²=0
B. Write the fundamental solutions for the associated homogeneous equation.
3₁ =
C. Write the form of the particular solution and its derivatives. (Use ABC etc. for undetermined
coefficients.
Y
-21
At³e-2t+Br²e-2t+Ct+D
Y' = A(-21³ +31²) e ²t +B(-2²+2t) e 2t+C
Y" = 4(4r³ — 12r² + 6t)e¯²² + B( 4t² — 8t+2) e¯²1
D. Write the general solution. (Use 1 and 2 for c₁ and c₂).
y
Y2 = te
y
E. Plug in the initial values and solve for c₁ and c₂ to find the solution to the initial value problem.
Hint: No fractions are required in the solution or answer to this problem.
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