Verify that

Answered question

2022-03-07

Verify that y1(x) = x^3 is a solution of x^2y''-5xy'+9y= 0. Hence find the general solution.

Answer & Explanation

star233

star233

Skilled2022-03-14Added 403 answers

Given ODE is

x2y5xy+9y=0 --- (i)

Since y1=x3 is a solution of above ODE then satisfies the equation (i) so,

x2y15xy1+9y=0            y1=x3x26x5x3x2+9x3            y1=3x26x315x3+9x3=0            y1=6x0=0

So, is a solution of given ODE.

Equation (i) is cauchy equation of Euler eqaution is a second order homogeneous linear equation.

Let z=lnx or x=ez and ddz=xddz0=xd
x2d2=0(01)xd=0

Then from eqaution (i) reduces to

0(01)y50y+9y=0(02050+0)y=0

(0260+9)y=0 --- (ii)

Auxiliary equation is

λ26λ+9=0(λ3)2=0λ=3,3

General solution of equation (ii) is,

y=(c1+c2z)e3z

Hence, the general solution of given ODE is,

y=(c1+c2lnx)x3

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