Explanation Given: The given differential equation is: t 2 x ″ + t x ′ + 16 x = 0 And the initial condition is: x ( 1 ) = 0 , x ′ ( 1 ) = 4 , t = 2 . Procedure used: Euler method: The initial value problem { d x d t = f ( t , x , y ) d x d t = f ( t , x , y ) ; x ( t 0 ) = x 0 , y ( t 0 ) = y 0 is approximated at each step by the recursive relationship based on the Taylor series expansion of x and y with step size h is, x n + 1 = x n + h f ( t n , x n , y n ) y n + 1 = y n + h g ( t n , x n , y n ) Here, t n = t 0 + n h , n = 0 , 1 , 2 , ... . The general solution of the linear homogeneous system X ′ = AX : First find the eigenvalues by using the formula | A − λ I | = 0 . A solution of λ to this equation gives eigenvalues and vector v satisfying ( A − λ I ) v = 0 is an eigenvectors corresponds to λ . The general solution is: X ( t ) = Φ ( t ) Φ − 1 ( 0 ) X 0 , where Φ ( t ) is the fundamental matrix. Calculation: Consider thedifferential equation: t 2 x ″ + t x ′ + 16 x = 0 Convert into first order equation: Let, x ′ = y then x ″ = y ′ . t 2 x ″ + t x ′ + 16 x = 0 t 2 y ′ + t y + 16 x = 0 y ′ = − t − 2 ( 16 x + t y ) The system of first order equations are: x ′ = y , y ′ = − t − 2 ( 16 x + t y ) , x ( 1 ) = 0 , y ( 1 ) = 4 Apply Euler’s method with h = 0.1 : x n + 1 = x n + 0.1 y n y n + 1 = y n + 0.1 ( − t n − 2 ( 16 x n + t n y n ) ) Find the value of x ( 2 ) and y ( 2 ) by creating the table x ( 1 ) = 0 , y ( 1 ) = 4 , t = 2 : n t n x n y n 0 1 0 4 1 1

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ISBN: 9780128149485

Author: Abell, Martha L. L.

Publisher: Elsevier Science

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Introductory Differential Equations

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