Question

We know that y1(x)=x3 is a solution to the differential equation x2D2y+5xDy−21y=0 forx∈(0,[infinity]).

Use the method of reduction of order to find a second solution tox2D2y+5xDy−21y=0 for x∈(0,[infinity]).(a) After you reduce the second order equation by making the substitution w=u′, you get a first order equation of the form w′=f(x,w)=(b) y2(X)=?

Answer 1

Explanation:

Given is a differntial equation
`x^2 y`,where x can take any positive value

One of the solution is

`y_1 = x^3`

Let us assume the second solution
`y_2 = u x^3`

Differentiate this y2 two times and plug in the DE to reduce the order

`y_2' = u'x^3 +3x^2 u\\y_2`

plug these in the DE

`u`

Put w=u'

xw'+11w=0

`y_2=ux^3`