Explanation Given data: The differential equation is, d 2 y d x 2 + d y d x − 2 y = x 2 y ″ + y ′ − 2 y = x 2 (1) Formula used: Write the expression for differential equation. a y ″ + b y ′ + c y = 0 (2) Write the expression for auxiliary equation. a r 2 + b r + c = 0 (3) Write the expression for general solution of a y ″ + b y ′ + c y = 0 with two distinct real roots. y c ( x ) = c 1 e r 1 x + c 2 e r 2 x (4) Here, r 1 and r 2 are the roots of auxiliary equation. Write the expression for the particular solution y p ( x ) . y p ( x ) = A x 2 + B x + C (5) Write the expression to find the roots of quadratic equation . r = − b ± b 2 − 4 a c 2 a (6) Compare equations (1) and (2). a = 1 b = 1 c = − 2 Substitute 1 for a , 1 for b , and –2 for c in equation (3), ( 1 ) r 2 + ( 1 ) r + ( − 2 ) = 0 r 2 + r − 2 = 0 Find the roots of auxiliary equation using equation (6). Substitute 1 for a , 1 for b , and –2 for c in equation (6), r = − ( 1 ) ± ( 1 ) 2 − 4 ( 1 ) ( − 2 ) 2 ( 1 ) = − 1 ± 1 + 8 2 = − 1 ± 9 2 = − 1 ± 3 2 Consider the value of r 1 and r 2 as follows. r 1 = − 1 + 3 2 = 2 2 = 1 r 2 = − 1 − 3 2 = − 2 Substitute 1 for r 1 and –2 for r 2 in equation (4), y c ( x ) = c 1 e x + c 2 e − 2 x (7) Differentiate equation (5) with respect to x , y ′ p ( x ) = d d x ( A x 2 + B x + C ) y ′ p ( x ) = 2 A x + B (8) Differentiate equation (8) with respect to x , y ″ p ( x ) = d d x ( 2 A x + B ) y ″ p ( x ) = 2 A (9) Substitute equations (5), (8) and (9) in equation (1), 2 A x + B + 2 A − 2 ( A x 2 + B x + C ) = x 2 2 A + 2 A x + B − 2 A x 2 − 2 B x − 2 C = x 2 (10) Substitute 0 for x in equation (10), 2 A + 2 A ( 0 ) + B − 2 A ( 0 )

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ISBN: 9781337771474

Author: Stewart

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Chapter 17, Problem 6RE

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The differential equation of d 2 y d x 2 + d y d x − 2 y = x 2 .

Solution Summary: The author explains the formula used to solve the differential equation.

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Chapter 17 Solutions