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Euler’s Differential Equation Euler’s differential equation is of the form x 2 y ″ + a x y ′ + b y = 0 , x > 0 Where a and b are constants. Show that this equation can be transformed into a second-order linear differential equation with constant coefficients by using the substitution x = e t . Solve x 2 y ″ + 6 x y ′ + 6 y = 0 .

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Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378
Chapter 16, Problem 8PS
Textbook Problem
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Euler’s Differential Equation Euler’s differential equation is of the form

x 2 y + a x y + b y = 0 , x > 0

Where a and b are constants.

Show that this equation can be transformed into a second-order linear differential equation with constant coefficients by using the substitution x = et.

Solve x 2 y + 6 x y + 6 y = 0 .

(a)

To determine

To prove: That the differential equation x2y+axy+by=0,x>0 can be transformed into a second-order linear differential equation with constant coefficients by using thesubstitution x=et.

Explanation of Solution

Given information:

The differential equation x2y+axy+by=0,x>0.

Proof:

As x=et.

Taking natural log on both the sides.

lnx=lnetlnx=t

Differentiate with respect to x.

dtdx=1x

Now, calculate the value of y.

y=dydt.dtdx=1x.dydt

Calculate the value of y.

y=ddx(dydx)=ddx(1x.dydt)=1x.ddx(dydt)+ddx(1x).dydt=1x.ddt(dydt)dtdx1x2.dydt=1x.ddt(dydt)1x1x2

(b)

To determine

To calculate: The solution of the equation x2y+6xy+6y=0.

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Chapter 16 Solutions

Multivariable Calculus
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