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LinearEquationsConstantCoeffieicnts

# LinearEquationsConstantCoeffieicnts - SECTION 17.5 Linear...

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SECTION 17.5: Linear Differential Equations with Constant CoefFcients 919 Exercises 17.4 1. Show that y = e x is a solution of y 00 3 y 0 + 2 y = 0, and Fnd the general solution of this DE. 2. Show that y = e 2 x is a solution of y 00 y 0 6 y = 0, and Fnd the general solution of this DE. 3. Show that y = x is a solution of x 2 y 00 + 2 xy 0 2 y = 0on the interval ( 0 , ) , and Fnd the general solution on this interval. 4. Show that y = x 2 is a solution of x 2 y 00 3 0 + 4 y = the interval ( 0 , ) , and Fnd the general solution on this interval. 5. Show that y = x is a solution of the differential equation x 2 y 00 ( 2 x + x 2 ) y 0 + ( 2 + x ) y = 0, and Fnd the general solution of this equation. 6. Show that y = x 1 / 2 cos x is a solution of the Bessel equation with ν = 1 / 2: x 2 y 00 + 0 + ± x 2 1 4 ² y = 0 . ±ind the general solution of this equation. First-order systems 7. Asystemof n Frst-order, linear, differential equations in n unknown functions y 1 , y 2 , ··· , y n is written y 0 1 = a 11 ( x ) y 1 + a 12 ( x ) y 2 +···+ a 1 n ( x ) y n + f 1 ( x ) y 0 2 = a 21 ( x ) y 1 + a 22 ( x ) y 2 a 2 n ( x ) y n + f 2 ( x ) . . . y 0 n = a n 1 ( x ) y 1 + a n 2 ( x ) y 2 a nn ( x ) y n + f n ( x ). Such a system is called an n × n frst-order linear system and can be rewritten in vector-matrix form as y 0 = A ( x ) y + ± ( x ) ,where y ( x ) = y 1 ( x ) . . . y n ( x ) , ± ( x ) = f 1 ( x ) . . . f n ( x ) , A ( x ) = a 11 ( x ) a 1 n ( x ) . . . . . . . . . a n 1 ( x ) a ( x ) . Show that the second-order, linear equation y 00 + a 1 ( x ) y 0 + a 0 ( x ) y = f ( x ) can be transformed into a 2 × 2 Frst-order system with y 1 = y and y 2 = y 0 having A ( x ) = ± 01 a 0 ( x ) a 1 ( x ) ² , ± ( x ) = ± 0 f ( x ) ² . 8. Generalize Exercise 7 to transform an n th-order linear equation y ( n ) + a n 1 ( x ) y ( n 1 ) + a n 2 ( x ) y ( n 2 ) a 0 ( x ) y = f ( x ) into an n × n Frst-order system. 9. If A is an n × n constant matrix, and if there exists a scalar λ and a nonzero constant vector v for which A v = λ v ,show that y = C 1 e λ x v is a solution of the homogeneous system y 0 = A y . 10. Show that the determinant ³ ³ ³ ³ 2 λ 1 23 λ ³ ³ ³ ³ is zero for two distinct values of λ . ±or each of these values Fnd a nonzero vector v that satisFes the condition ± 21 ² v = λ v .Hence solve the system y 0 1 = 2 y 1 + y 2 , y 0 2 = 2 y 1 + 3 y 2 . 17.5 Linear Differential Equations with Constant CoefFcients A differential equation of the form ay 00 + by 0 + cy = 0 ,( ) where a , b ,and c are constants and a 6= 0, is said to be a linear, homogeneous, second-order equation with constant coe±fcients . A thorough discussion o± techniques ±or solving such equations, together with examples, exercises, and applications to the study o± simple and damped harmonic motion, can be ±ound in Section 3.7; we will not repeat that discussion here. I± you have not studied it, please do so now.

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LinearEquationsConstantCoeffieicnts - SECTION 17.5 Linear...

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