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A general solution of the differential equation ( D 2 + 2 D + 2 I ) y = 3 e − x cos 2 x . The solution of the differential equation is y = e − x [ ( − cos 2 x ) + ( c 1 cos x + c 2 sin x ) ] _ . The characteristics equation of the homogeneous differential equation is written as: ( D 2 + 2 D + 2 I ) y = 0 D ( D y ) + 2 D y + 2 y I = 0 y ″ + 2 y ′ + 2 y = 0 λ 2 + 2 λ + 2 = 0 On further simplification, the following is obtained. λ = − 2 ± 4 − 4 × 2 2 λ = − 2 ± − 4 2 λ = − 1 ± 2 i So, the general solution of the ordinary differential equations is y h = e − x ( c 1 cos x + c 2 sin x ) . Now the particular solution of the differential equation is calculated as: y p = e − x ( K cos 2 x + M sin 2 x ) Differentiate the above solution with respect to x . y ′ p = d d x [ e − x ( K cos 2 x + M sin 2 x ) ] = − e − x ( K cos 2 x + M sin 2 x ) − 2 K e − x sin 2 x + 2 M e − x cos 2 x = e − x [ ( − 2 K − M ) sin 2 x + ( − K + 2 M ) cos 2 x ] = e − x [ ( 2 M − K ) cos 2 x − ( 2 K + M ) sin 2 x ] Differentiate the above function with respect to x . y ″ p = d d x [ e − x [ ( 2 M − K ) cos 2 x − ( 2 K + M ) sin 2 x ] ] = − e − x [ ( 2 M − K ) cos 2 x − ( 2 K + M ) sin 2 x ] + e − x [ − 2 ( 2 M − K ) sin 2 x − 2 ( 2 K + M ) cos 2 x ] = e − x [ K − 2 M − 4 K − 2 M ] cos ( 2 x ) + ( 2 K + M + 2 K − 4 M ) sin 2 x = e − x [ − 3 K − 4 M ] cos ( 2 x ) + ( 4 K − 3 M ) sin 2 x Substitute, the value of y ″ p and y ′ p in the equation y ″ + 2 y ′ + 2 y = 3 e − x cos 2 x ( [ e − x [ − 3 K − 4 M ] cos ( 2 x ) + ( 4 K − 3 M ) sin 2 x ] + 2 [ e − x [ ( 2 M − K ) cos 2 x − ( 2 K + M ) sin 2 x ] ] + 2 [ e − x ( K cos 2 x + M sin 2 x ) ] ) = 3 e − x cos 2 x ( e − x [ − 3 K − 4 M ] cos ( 2 x ) + ( 4 K − 3 M ) sin 2 x + ( 4 K − 3 M ) cos 2 x + ( 4 K − 2 M ) cos 2 x − ( 4 K + 2 M ) sin 2 x + 2 K cos 2 x + 2 M cos 2 x ) = 3 e − x cos 2 x e − x [ − 3 K − 4 M + 4 M − 2 K ] sin 2 x + [ 2 M − 4 K − 2 M + 4 K − 3 K ] cos 2 x = 3 e − x cos 2 x − 3 M e − x sin 2 x − 3 K e − x cos 2 x = 3 e − x cos 2 x Comparing the coefficients on both sides of the above equation. M = 0 K = − 1 Substitute, M = 0 and K = − 1 in equation y p = e − x ( K cos 2 x + M sin 2 x ) . y p = e − x ( − 1 × cos 2 x + 0 × sin 2 x ) y p = − e − x ( cos 2 x ) So, the solution of the differential equation is calculated as: y = y p + y h = e − x ( − cos 2 x ) + e − x ( c 1 cos x + c 2 sin x ) = e − x [ ( − cos 2 x ) + ( c 1 cos x + c 2 sin x ) ] Therefore, the solution of the differential equation is y = e − x [ ( − cos 2 x ) + ( c 1 cos x + c 2 sin x ) ] _ .

A general solution of the differential equation ( D 2 + 2 D + 2 I ) y = 3 e − x cos 2 x . The solution of the differential equation is y = e − x [ ( − cos 2 x ) + ( c 1 cos x + c 2 sin x ) ] _ . The characteristics equation of the homogeneous differential equation is written as: ( D 2 + 2 D + 2 I ) y = 0 D ( D y ) + 2 D y + 2 y I = 0 y ″ + 2 y ′ + 2 y = 0 λ 2 + 2 λ + 2 = 0 On further simplification, the following is obtained. λ = − 2 ± 4 − 4 × 2 2 λ = − 2 ± − 4 2 λ = − 1 ± 2 i So, the general solution of the ordinary differential equations is y h = e − x ( c 1 cos x + c 2 sin x ) . Now the particular solution of the differential equation is calculated as: y p = e − x ( K cos 2 x + M sin 2 x ) Differentiate the above solution with respect to x . y ′ p = d d x [ e − x ( K cos 2 x + M sin 2 x ) ] = − e − x ( K cos 2 x + M sin 2 x ) − 2 K e − x sin 2 x + 2 M e − x cos 2 x = e − x [ ( − 2 K − M ) sin 2 x + ( − K + 2 M ) cos 2 x ] = e − x [ ( 2 M − K ) cos 2 x − ( 2 K + M ) sin 2 x ] Differentiate the above function with respect to x . y ″ p = d d x [ e − x [ ( 2 M − K ) cos 2 x − ( 2 K + M ) sin 2 x ] ] = − e − x [ ( 2 M − K ) cos 2 x − ( 2 K + M ) sin 2 x ] + e − x [ − 2 ( 2 M − K ) sin 2 x − 2 ( 2 K + M ) cos 2 x ] = e − x [ K − 2 M − 4 K − 2 M ] cos ( 2 x ) + ( 2 K + M + 2 K − 4 M ) sin 2 x = e − x [ − 3 K − 4 M ] cos ( 2 x ) + ( 4 K − 3 M ) sin 2 x Substitute, the value of y ″ p and y ′ p in the equation y ″ + 2 y ′ + 2 y = 3 e − x cos 2 x ( [ e − x [ − 3 K − 4 M ] cos ( 2 x ) + ( 4 K − 3 M ) sin 2 x ] + 2 [ e − x [ ( 2 M − K ) cos 2 x − ( 2 K + M ) sin 2 x ] ] + 2 [ e − x ( K cos 2 x + M sin 2 x ) ] ) = 3 e − x cos 2 x ( e − x [ − 3 K − 4 M ] cos ( 2 x ) + ( 4 K − 3 M ) sin 2 x + ( 4 K − 3 M ) cos 2 x + ( 4 K − 2 M ) cos 2 x − ( 4 K + 2 M ) sin 2 x + 2 K cos 2 x + 2 M cos 2 x ) = 3 e − x cos 2 x e − x [ − 3 K − 4 M + 4 M − 2 K ] sin 2 x + [ 2 M − 4 K − 2 M + 4 K − 3 K ] cos 2 x = 3 e − x cos 2 x − 3 M e − x sin 2 x − 3 K e − x cos 2 x = 3 e − x cos 2 x Comparing the coefficients on both sides of the above equation. M = 0 K = − 1 Substitute, M = 0 and K = − 1 in equation y p = e − x ( K cos 2 x + M sin 2 x ) . y p = e − x ( − 1 × cos 2 x + 0 × sin 2 x ) y p = − e − x ( cos 2 x ) So, the solution of the differential equation is calculated as: y = y p + y h = e − x ( − cos 2 x ) + e − x ( c 1 cos x + c 2 sin x ) = e − x [ ( − cos 2 x ) + ( c 1 cos x + c 2 sin x ) ] Therefore, the solution of the differential equation is y = e − x [ ( − cos 2 x ) + ( c 1 cos x + c 2 sin x ) ] _ .

ADVANCED ENGINEERING MATH.>CUSTOM<

ADVANCED ENGINEERING MATH.>CUSTOM<

10th Edition

ISBN: 9781119480150

Author: Kreyszig

Publisher: WILEY C

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Chapter 2 Solutions

ADVANCED ENGINEERING MATH.>CUSTOM<

Ch. 2.1 - Prob. 1P Ch. 2.1 - Reduction. Show that F(y, y′, y″) = 0 can be...Ch. 2.1 - 3–10 REDUCTION OF ORDER Reduce to first order and...Ch. 2.1 - 3–10 REDUCTION OF ORDER Reduce to first order and...Ch. 2.1 - 3–10 REDUCTION OF ORDER Reduce to first order and...Ch. 2.1 - 3–10 REDUCTION OF ORDER Reduce to first order and...Ch. 2.1 - 3–10 REDUCTION OF ORDER Reduce to first order and...Ch. 2.1 - 3–10 REDUCTION OF ORDER Reduce to first order and...Ch. 2.1 - 3–10 REDUCTION OF ORDER Reduce to first order and...Ch. 2.1 - 3–10 REDUCTION OF ORDER Reduce to first order and...

Ch. 2.1 - 11–14 APPLICATIONS OF REDUCIBLE...Ch. 2.1 - 11–14 APPLICATIONS OF REDUCIBLE ODEs 12. Hanging...Ch. 2.1 - APPLICATIONS OF REDUCIBLE ODEs 13. Motion. If, in...Ch. 2.1 - Motion. In a straight-line motion, let the...Ch. 2.1 - GENERAL SOLUTION. INITIAL VALUE PROBLEM...Ch. 2.1 - GENERAL SOLUTION. INITIAL VALUE PROBLEM...Ch. 2.1 - GENERAL SOLUTION. INITIAL VALUE PROBLEM...Ch. 2.1 - GENERAL SOLUTION. INITIAL VALUE PROBLEM...Ch. 2.1 - GENERAL SOLUTION. INITIAL VALUE PROBLEM...Ch. 2.2 - 1–15 GENERAL SOLUTION Find a general solution....Ch. 2.2 - 1–15 GENERAL SOLUTION Find a general solution....Ch. 2.2 - 1–15 GENERAL SOLUTION Find a general solution....Ch. 2.2 - Find a general solution. Check your answer by...Ch. 2.2 - Find a general solution. Check your answer by...Ch. 2.2 - Find a general solution. Check your answer by...Ch. 2.2 - 1–15 GENERAL SOLUTION Find a general solution....Ch. 2.2 - Find a general solution. Check your answer by...Ch. 2.2 - 1–15 GENERAL SOLUTION Find a general solution....Ch. 2.2 - GENERAL SOLUTION Find a general solution. Check...Ch. 2.2 - 1–15 GENERAL SOLUTION Find a general solution....Ch. 2.2 - 1–15 GENERAL SOLUTION Find a general solution....Ch. 2.2 - Find a general solution. Check your answer by...Ch. 2.2 - Find a general solution. Check your answer by...Ch. 2.2 - Find a general solution. Check your answer by...Ch. 2.2 - 16–20 FIND AN ODE for the given basis. 16. Ch. 2.2 - 16–20 FIND AN ODE for the given basis. 17. Ch. 2.2 - 16–20 FIND AN ODE for the given basis. 18. Ch. 2.2 - 16–20 FIND AN ODE for the given basis. 19. Ch. 2.2 - 16–20 FIND AN ODE for the given basis. 20. Ch. 2.2 - Solve the IVP. Check that your answer satisfies...Ch. 2.2 - Solve the IVP. Check that your answer satisfies...Ch. 2.2 - Solve the IVP. Check that your answer satisfies...Ch. 2.2 - Solve the IVP. Check that your answer satisfies...Ch. 2.2 - Solve the IVP. Check that your answer satisfies...Ch. 2.2 - Solve the IVP. Check that your answer satisfies...Ch. 2.2 - Solve the IVP. Check that your answer satisfies...Ch. 2.2 - Solve the IVP. Check that your answer satisfies...Ch. 2.2 - INITIAL VALUES PROBLEMS Solve the IVP. Check that...Ch. 2.2 - Solve the IVP. Check that your answer satisfies...Ch. 2.2 - LINEAR INDEPENDENCE is of basic importance, in...Ch. 2.2 - LINEAR INDEPENDENCE is of basic importance, in...Ch. 2.2 - Prob. 33P Ch. 2.2 - Prob. 34P Ch. 2.2 - Prob. 35P Ch. 2.2 - LINEAR INDEPENDENCE is of basic importance, in...Ch. 2.2 - Instability. Solve y″ − y = 0 for the initial...Ch. 2.3 - Apply the given operator to the given functions....Ch. 2.3 - Apply the given operator to the given functions....Ch. 2.3 - Apply the given operator to the given functions....Ch. 2.3 - Prob. 4P Ch. 2.3 - Apply the given operator to the given functions....Ch. 2.3 - Factor as in the text and solve. (D2 + 4.00D +...Ch. 2.3 - Factor as in the text and solve. (4D2 − I)y = 0 Ch. 2.3 - Factor as in the text and solve. (D2 + 3I)y = 0 Ch. 2.3 - Factor as in the text and solve. (D2 − 4.20D +...Ch. 2.3 - Factor as in the text and solve. (D2 + 4.80D +...Ch. 2.3 - Factor as in the text and solve. (D2 − 4.00D +...Ch. 2.3 - Prob. 12P Ch. 2.3 - Linear operator. Illustrate the linearity of L in...Ch. 2.3 - Double root. If D2 + aD + bI has distinct roots μ...Ch. 2.3 - Definition of linearity. Show that the definition...Ch. 2.4 - Initial value problem. Find the harmonic motion...Ch. 2.4 - Frequency. If a weight of 20 nt (about 4.5 lb)...Ch. 2.4 - Frequency. How does the frequency of the harmonic...Ch. 2.4 - Initial velocity. Could you make a harmonic...Ch. 2.4 - Springs in parallel. What are the frequencies of...Ch. 2.4 - Spring in series. If a body hangs on a spring s1...Ch. 2.4 - Prob. 7P Ch. 2.4 - Prob. 8P Ch. 2.4 - HARMONIC OSCILLATIONS (UNDAMPED MOTION) 9....Ch. 2.4 - Prob. 11P Ch. 2.4 - DAMPED MOTION 12. Overdamping. Show that in the...Ch. 2.4 - DAMPED MOTION 13. Initial value problem. Find the...Ch. 2.4 - DAMPED MOTION 14. Shock absorber. What is the...Ch. 2.4 - DAMPED MOTION 15. Frequency. Find an approximation...Ch. 2.4 - DAMPED MOTION 16. Maxima. Show that the maxima of...Ch. 2.4 - DAMPED MOTION 17. Underdamping. Determine the...Ch. 2.4 - DAMPED MOTION 18. Logarithmic decrement. Show that...Ch. 2.4 - DAMPED MOTION 19. Damping constant. Consider an...Ch. 2.5 - Prob. 1P Ch. 2.5 - Find a real general solution. Show the details of...Ch. 2.5 - Find a real general solution. Show the details of...Ch. 2.5 - Find a real general solution. Show the details of...Ch. 2.5 - Find a real general solution. Show the details of...Ch. 2.5 - Find a real general solution. Show the details of...Ch. 2.5 - Find a real general solution. Show the details of...Ch. 2.5 - Find a real general solution. Show the details of...Ch. 2.5 - Prob. 9P Ch. 2.5 - Find a real general solution. Show the details of...Ch. 2.5 - Find a real general solution. Show the details of...Ch. 2.5 - INITIAL VALUE PROBLEM Solve and graph the...Ch. 2.5 - INITIAL VALUE PROBLEM Solve and graph the...Ch. 2.5 - INITIAL VALUE PROBLEM Solve and graph the...Ch. 2.5 - INITIAL VALUE PROBLEM Solve and graph the...Ch. 2.5 - INITIAL VALUE PROBLEM Solve and graph the...Ch. 2.5 - INITIAL VALUE PROBLEM Solve and graph the...Ch. 2.5 - INITIAL VALUE PROBLEM Solve and graph the...Ch. 2.5 - INITIAL VALUE PROBLEM Solve and graph the...Ch. 2.6 - Derive (6*) from (6). Ch. 2.6 - BASIS OF SOLUTIONS. WRONSKIAN Find the Wronskian....Ch. 2.6 - BASIS OF SOLUTIONS. WRONSKIAN Find the Wronskian....Ch. 2.6 - BASIS OF SOLUTIONS. WRONSKIAN Find the Wronskian....Ch. 2.6 - BASIS OF SOLUTIONS. WRONSKIAN Find the Wronskian....Ch. 2.6 - BASIS OF SOLUTIONS. WRONSKIAN Find the Wronskian....Ch. 2.6 - BASIS OF SOLUTIONS. WRONSKIAN Find the Wronskian....Ch. 2.6 - BASIS OF SOLUTIONS. WRONSKIAN Find the Wronskian....Ch. 2.6 - Prob. 9P Ch. 2.6 - ODE FOR GIVEN BASIS. WRONSKIAN. IVP (a) Find a...Ch. 2.6 - ODE FOR GIVEN BASIS. WRONSKIAN. IVP (a) Find a...Ch. 2.6 - ODE FOR GIVEN BASIS. WRONSKIAN. IVP (a) Find a...Ch. 2.6 - ODE FOR GIVEN BASIS. WRONSKIAN. IVP (a) Find a...Ch. 2.6 - ODE FOR GIVEN BASIS. WRONSKIAN. IVP (a) Find a...Ch. 2.6 - ODE FOR GIVEN BASIS. WRONSKIAN. IVP (a) Find a...Ch. 2.7 - NONHOMOGENEOUS LINEAR ODEs: GENERAL SOLUTION Find...Ch. 2.7 - NONHOMOGENEOUS LINEAR ODEs: GENERAL SOLUTION Find...Ch. 2.7 - NONHOMOGENEOUS LINEAR ODEs: GENERAL SOLUTION Find...Ch. 2.7 - NONHOMOGENEOUS LINEAR ODEs: GENERAL SOLUTION Find...Ch. 2.7 - NONHOMOGENEOUS LINEAR ODEs: GENERAL SOLUTION Find...Ch. 2.7 - NONHOMOGENEOUS LINEAR ODEs: GENERAL SOLUTION Find...Ch. 2.7 - NONHOMOGENEOUS LINEAR ODEs: GENERAL SOLUTION Find...Ch. 2.7 - NONHOMOGENEOUS LINEAR ODEs: GENERAL SOLUTION Find...Ch. 2.7 - NONHOMOGENEOUS LINEAR ODEs: GENERAL SOLUTION Find...Ch. 2.7 - NONHOMOGENEOUS LINEAR ODEs: GENERAL SOLUTION Find...Ch. 2.7 - NONHOMOGENEOUS LINEAR ODEs: IVPs Solve the initial...Ch. 2.7 - NONHOMOGENEOUS LINEAR ODEs: IVPs Solve the initial...Ch. 2.7 - NONHOMOGENEOUS LINEAR ODEs: IVPs Solve the initial...Ch. 2.7 - NONHOMOGENEOUS LINEAR ODEs: IVPs Solve the initial...Ch. 2.7 - NONHOMOGENEOUS LINEAR ODEs: IVPs Solve the initial...Ch. 2.7 - NONHOMOGENEOUS LINEAR ODEs: IVPs Solve the initial...Ch. 2.7 - NONHOMOGENEOUS LINEAR ODEs: IVPs Solve the initial...Ch. 2.7 - NONHOMOGENEOUS LINEAR ODEs: IVPs Solve the initial...Ch. 2.7 - CAS PROJECT. Structure of Solutions of Initial...Ch. 2.8 - Prob. 2P Ch. 2.8 - Find the steady-state motion of the mass–spring...Ch. 2.8 - Find the steady-state motion of the mass–spring...Ch. 2.8 - Find the steady-state motion of the mass–spring...Ch. 2.8 - Prob. 6P Ch. 2.8 - Prob. 7P Ch. 2.8 - TRANSIENT SOLUTIONS Find the transient motion of...Ch. 2.8 - TRANSIENT SOLUTIONS Find the transient motion of...Ch. 2.8 - TRANSIENT SOLUTIONS Find the transient motion of...Ch. 2.8 - TRANSIENT SOLUTIONS Find the transient motion of...Ch. 2.8 - Prob. 12P Ch. 2.8 - Prob. 13P Ch. 2.8 - TRANSIENT SOLUTIONS Find the transient motion of...Ch. 2.8 - TRANSIENT SOLUTIONS Find the transient motion of...Ch. 2.8 - INITIAL VALUE PROBLEMS Find the motion of the...Ch. 2.8 - Prob. 17P Ch. 2.8 - INITIAL VALUE PROBLEMS Find the motion of the...Ch. 2.8 - Prob. 19P Ch. 2.8 - Prob. 20P Ch. 2.8 - Prob. 21P Ch. 2.8 - Prob. 22P Ch. 2.8 - Prob. 24P Ch. 2.9 - RC-Circuit. Model the RC-circuit in Fig. 64. Find...Ch. 2.9 - RC-Circuit. Solve Prob. 1 when E = E0 sin ωt and...Ch. 2.9 - RL-Circuit. Model the RL-circuit in Fig. 66. Find...Ch. 2.9 - RL-Circuit. Solve Prob. 3 when E = E0 sin ωt and...Ch. 2.9 - LC-Circuit. This is an RLC-circuit with negligibly...Ch. 2.9 - LC-Circuit. Find the current when L = 0.5 H, C =...Ch. 2.9 - Prob. 7P Ch. 2.9 - 8–14 Find the steady-state current in the...Ch. 2.9 - 8–14 Find the steady-state current in the...Ch. 2.9 - 8–14 Find the steady-state current in the...Ch. 2.9 - 8–14 Find the steady-state current in the...Ch. 2.9 - Find the steady-state current in the RLC-circuit...Ch. 2.9 - Find the steady-state current in the RLC-circuit...Ch. 2.9 - Prob. 14P Ch. 2.9 - Prob. 15P Ch. 2.9 - Solve the initial value problem for the...Ch. 2.9 - Prob. 17P Ch. 2.9 - Prob. 18P Ch. 2.9 - Complex Solution Method. Solve , by substituting...Ch. 2.10 - Solve the given nonhomogeneous linear ODE by...Ch. 2.10 - Solve the given nonhomogeneous linear ODE by...Ch. 2.10 - Solve the given nonhomogeneous linear ODE by...Ch. 2.10 - Solve the given nonhomogeneous linear ODE by...Ch. 2.10 - Prob. 5P Ch. 2.10 - Solve the given nonhomogeneous linear ODE by...Ch. 2.10 - Solve the given nonhomogeneous linear ODE by...Ch. 2.10 - Solve the given nonhomogeneous linear ODE by...Ch. 2.10 - Solve the given nonhomogeneous linear ODE by...Ch. 2.10 - Solve the given nonhomogeneous linear ODE by...Ch. 2.10 - Solve the given nonhomogeneous linear ODE by...Ch. 2.10 - Prob. 12P Ch. 2.10 - Solve the given nonhomogeneous linear ODE by...Ch. 2 - Prob. 1RQ Ch. 2 - Prob. 2RQ Ch. 2 - By what methods can you get a general solution of...Ch. 2 - Prob. 4RQ Ch. 2 - Prob. 5RQ Ch. 2 - Prob. 6RQ Ch. 2 - Find a general solution. Show the details of your...Ch. 2 - Find a general solution. Show the details of your...Ch. 2 - Find a general solution. Show the details of your...Ch. 2 - Find a general solution. Show the details of your...Ch. 2 - Find a general solution. Show the details of your...Ch. 2 - Find a general solution. Show the details of your...Ch. 2 - Find a general solution. Show the details of your...Ch. 2 - Find a general solution. Show the details of your...Ch. 2 - Prob. 15RQ Ch. 2 - Prob. 16RQ Ch. 2 - Prob. 17RQ Ch. 2 - Find a general solution. Show the details of your...Ch. 2 - Solve the problem, showing the details of your...Ch. 2 - Solve the problem, showing the details of your...Ch. 2 - Solve the problem, showing the details of your...Ch. 2 - Solve the problem, showing the details of your...Ch. 2 - Find the steady-state current in the RLC-circuit...Ch. 2 - Find a general solution of the homogeneous linear...Ch. 2 - Find the steady-state current in the RLC-circuit...Ch. 2 - Find the current in the RLC-circuit in Fig. 71...Ch. 2 - Prob. 27RQ Ch. 2 - Prob. 28RQ Ch. 2 - Prob. 29RQ Ch. 2 - Prob. 30RQ

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