Explanation Given: The given differential equation is y ( 4 ) + y = ( 16 x 4 − 11 ) sin ( x 2 ) − 48 x 2 cos ( x 2 ) . The particular solution to the above differential equation is y p = sin ( x 2 ) . Approach: The general solution to the differential equation a n y ( n ) + a n − 1 y ( n − 1 ) + ⋅ ⋅ ⋅ + a 0 y = 0 is obtained The characteristic equation to the differential equation is given by a n r n + a n − 1 r n − 1 + ⋅ ⋅ ⋅ + a 0 = 0 • If there are n distinct roots of the characteristic equation, then the solution set is { e r 1 x , e r 2 x , ... , e r n x } . If some real roots occurs with multiplicity m , then m of the functions are replaced by, e r 1 x , x e r 1 x , ... , x m − 1 e r 1 x . • If a complex root α + i β to the characteristic equation occurs with multiplicity m , then the solution is given as, e α x sin β x , x e α x sin β x , ... , x m − 1 e α x sin β x Or, e α x cos β x , x e α x cos β x , ... , x m − 1 e α x cos β x Therefore, a general solution to the non homogeneous equation is given as, y ( x ) = y h ( x ) + y p ( x ) . Calculation: Consider the given differential equation, y ( 4 ) + y = ( 16 x 4 − 11 ) sin ( x 2 ) − 48 x 2 cos ( x 2 ) . It’s characteristic equation is given as, r 4 + 1 = 0 Solve the characteristic equation for r as follows, r 4 = − 1 Rewrite the equation with u = r 2 and u 2 = r 4 , thus, u 2 = − 1 u 2 = i 2 u = ± i As, u = r 2 , solve the equations u 2 = i and u 2 = − i in order to find r .

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Solution Summary: The author explains the general solution to the given differential equation: y(4)+y=(16x4-11)mathrmsin

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To find:

The general solution to the given differential equation.