Explanation Given: The differential equation is y ″ ( θ ) + 4 y ′ ( θ ) + 7 y ( θ ) = 2 cos θ . The initial value conditions are y ( 0 ) = 3 , y ′ ( 0 ) = 0 Approach: The auxiliary equation of a differential equation a n y ( n ) + a n − 1 y ( n − 1 ) + ⋅ ⋅ ⋅ + a 1 y ( 1 ) + a 0 is a n r n + a n − 1 r n − 1 + ⋅ ⋅ ⋅ + a 1 r 1 + a 0 ... ( 1 ) . Here, y ( n ) denotes the n th derivative of y . If x = a is a zero of the polynomial a n x n + a n − 1 x n − 1 + ⋅ ⋅ ⋅ + a 1 x 1 + a 0 , then ( x − a ) is a factor of the polynomial ... ( 2 ) . If the auxiliary equation has roots r 1 = a + b i , r 2 = a − b i then the solution of the differential equation is y = A e r 1 t + B e r 2 t ... ( 3 ) If the auxiliary equation has same root r 1 = r 2 = r , then the solution of the differential equation is y = A e r t + t B e r t Calculation: The differential equation is y ″ ( θ ) + 2 y ′ ( θ ) + y ( θ ) = 2 cos θ . The auxiliary equation of the differential equation is r 2 + 2 r + 1 = 0 Let p ( r ) = r 2 + 2 r + 1 Put p ( r ) = 0 r 2 + 2 r + 1 = 0 r 2 + r + r + 1 = 0 r ( r + 1 ) + r ( r + 1 ) = 0 ( r + 1 ) 2 = 0 The roots are − 1 , − 1 . y h ( θ ) = C 1 e − θ + C 2 θ e − θ ( 4 ) . p ( θ ) = 2 cos θ . The particular solution is the linear combination of 2 cos θ and its possible derivatives. y p ( θ ) = A cos θ + B sin θ . Differentiate y p ( t ) 2 times with respect to t . y ′ p ( θ ) = − A sin θ + B cos θ y ″ p ( θ ) = − A cos θ − B sin θ Since, y p is a solution of y ″ ( θ ) + 2 y ′ ( θ ) + y ( θ ) = 2 cos θ By ( 2 ) , y ″ p + 2 y ′ p + y p = 2 cos θ Substitute A cos θ + B sin θ for y p , − A sin θ + B cos θ for y ′ p , − A cos θ − B sin θ for y ″ p in y ″ p + 2 y ′ p + y p = 2 cos θ

Pearson eText Fundamentals of Differential Equations with Boundary Value Problems -- Instant Access (Pearson+)

7th Edition

ISBN: 9780137394524

Author: R. Nagle, Edward Saff

Publisher: PEARSON+

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Pearson eText Fundamentals of Differential Equations with Boundary Value Problems -- Instant Access (Pearson+)

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The solution to the differential equation

Solution Summary: The author explains the solution of the differential equation: y(theta )+4y

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The solution to the differential equation