To determine To solve: The ODE y ″ + 25 y = 0 for the given initial values and check whether the answer satisfies the ODE also the initial conditions. Explanation Result used: Case (i): If the auxiliary equation has distinct real roots λ 1 , and λ 2 of y = e λ x , then the basis of y ″ + a y ′ + b y = 0 is e λ 1 x , e λ 2 x and the general solution of y ″ + a y ′ + b y = 0 is y = c 1 e λ 1 x + c 2 e λ 2 x . Case (ii): If the auxiliary equation has real double root λ = − a of y = e λ x , then the basis of y ″ + a y ′ + b y = 0 is e − a x , x e − a x and the general solution of y ″ + a y ′ + b y = 0 is y = ( c 1 + c 2 ) e − a x . Case (iii): If the auxiliary equation has complex conjugate root λ 1 = a + i ω , λ 2 = a − i ω of y = e λ x , then the basis of y ″ + a y ′ + b y = 0 is e a x cos ω x , e a x sin ω x and the general solution of y ″ + a y ′ + b y = 0 is y = e − a x ( A cos ω x + B sin ω x ) . Calculation: The given ODE is y ″ + 25 y = 0 (1) Compute the auxiliary equation of the above ODE as follows. Let y = e λ x . That implies, the first and second derivative of y is y ′ = λ e λ x and y ″ = λ 2 e λ x , respectively. Substitute y ″ = λ 2 e λ x and y = e λ x in (1) and simplify further. λ 2 e λ x + 25 e λ x = 0 e λ x ( λ 2 + 25 ) = 0 λ 2 + 25 = 0 That is, the characteristic equation is λ 2 + 25 = 0 . Compute the roots of the above characteristic equation as follows. λ 2 + 25 = 0 λ 2 = − 25 λ = ± − 25 λ = ± 5 i From the above calculation, it is observed that y ″ + 25 y = 0 has complex conjugate roots. It is known that, the general form of the complex number is a + i b . Here, a = 0 and b = 5. Then by case (iii), the basis of y ″ + 25 y = 0 is e 0 ⋅ x cos 5 x , e 0 ⋅ x sin 5 x . The general solution of y ″ + 25 y = 0 is y = c 1 e 0 ⋅ x cos 5 x + c 2 e 0 ⋅ x sin 5 x , that is, y = c 1 cos 5 x + c 2 sin 5 x

10th Edition

ISBN: 9780470458365

Author: Erwin Kreyszig

Publisher: Wiley, John & Sons, Incorporated

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

100%

Chapter 2.2, Problem 21P

To determine

To solve: The ODE y″+25y=0 for the given initial values and check whether the answer satisfies the ODE also the initial conditions.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 2.2, Problem 20P

Chapter 2.2, Problem 22P

Students have asked these similar questions

read chapter 10 welding principles and applications, short-circuiting, transfer, GMAW – S. Explain why the power supply is critical in the short-circuiting transfer process. Discuss what happens if there’s too much or two little inductance.

a. Find the general flow pattern of the network shown in the figure. b. Assuming that the flow must be in the directions indicated, find the minimum flows in the branches denoted by X2, X3, X4, and x5 a. Choose the correct answer below and fill in the answer boxes to complete your choice. OA. x₁ = X2 is free X3 is free B. X₁ is free x2= ×4 is free X5 is free X6 = X3= X4 X5 X6 = 11 = ○ C. D. X2 is free X3= X4 is free X5 is free x2 = 0 X3 is free ×4 = X6 is free gave 20 30 12 C 804 60-> B <<90 314 X4D -80 E T 20 40 x

Consider an economy with three sectors, Chemicals & Metals, Fuels & Power, and Machinery. Chemicals sells 30% of its output to Fuels and 60% to Machinery and retains the rest. Fuels sells 70% of its output to Chemicals and 20% to Machinery and retains the rest. Machinery sells 40% of its output to Chemicals and 30% to Fuels and retains the rest. Complete parts (a) through (c) below a. Construct the exchange table for this economy. Distribution of Output from: Chemicals Fuels Machinery (Type integers or decimals.) Purchased by: Chemicals Fuels Machinery

Chapter 2 Solutions

Advanced Engineering Mathematics

Knowledge Booster

$Advanced Math$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.