15Consider the initial value problem y' -- y = 5t + 4e¹, y(0) = yo.2Find the value of yo that separates solutions that grow positively ast → ∞ from those that grow negatively. How does the solution thatcorresponds to this critical value of yo behave as t → ∞o?YoAs t → ∞o, the solution that corresponds to this critical value of yoChoose one

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Answered: 15 Consider the initial value 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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16. Study the stability of the trivial solution of x" + 5x" + 9x' + 5x = 0.
14)
Q2. (a) If b2 – 4ac = 0 and y1 = e"ma solves ay" + by' + cy = 0, use reduction of order to get a linearly independent solution. [You may assume linear independence]
What geometric transformation of 3D space happens in this matrix-vector multiplication? 010 -1 0 0 0 01 9).( (3) Please note: All rotation directions to be understood as when looking along the rotation axis towards the origin. O Clockwise rotation by 90 degrees around the x3 axis Counter-clockwise rotation by 90 degrees around the x3 axis Clockwise rotation by 90 degrees around the x₂ axis Counter-clockwise rotation by 90 degrees around the x2 axis O Clockwise rotation by 90 degrees around the x₁ axis Counter-clockwise rotation by 90 degrees around the x₁ axis O Mirroring across the x₁-x2 plane Mirroring across the x₁-x3 plane Mirroring across the X₂-X3 plane
6. Determine the General Solution of this Equation in terms of C += xy? xy 3DC y'-x+ b. y2+x-e2x 2. 3DC y"- y'+x+e** a. None of the above 2.
Show every step, and every rule. y 3 ((1)/x^2+C)) is a one-parameter siblings of solutions of: y' + 2xy2 = 0. Look for a simple solution of the DE that satisfies the initial condition: y(2)= 3
Verify that if y₁ and y2 are solutions of y" + p(t)y' + q(t)y y = C₁Y₁ + C2Y2 for any constants C₁ and C₂. = 0, then so is
3. Find the general solution of the following DE. y' = 3 (y + 1) O y = ce³x - 1 3y = ce³x - 1 y = 3ce³x - 1 O y = ce³x - 3/
9. Let A= 3 3 6 Sketch the solutions curves for ()-^) with initial values (-1,0), (1, 1), (1, 0). Your sketch should show the asymptotic behavior of the solutions when t→→ +∞o and t-o. On the trajectory, indicate the direction for t→+oo with an arrow. Also compute e¹A, 420
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4. Find the solution of the IVP y" - 4/ — 12y = 3e", y(0) = 18 =-=- y(0) 7,
Find the general solution of the following DES. 1. y' + 2y = x²y³ 2. y' — 2x³y = x³y=¹ 3. y' + y = x²y-² 4. y' + x³y = x³y-2 h

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