Q18E

Question

Suppose r₀ is a repeated root of the auxiliary equation ar²+br+c=0. Then, as we well know, is a solution to the equation ay"+by'+cy=0 where a, b, and c are constants. Use a derivation similar to the one given in this section for the case when the indicial equation has a repeated root to show that a second linearly independent solution is y₂ (t)=te^rt .

Step-by-Step Solution

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Answer

The second solution for the differential equation is derived to be y₂=te^r₀^t where the first solution is given to be y₁=e^r₀^t .

1Definition of Cauchy-Euler equation:

It has direct application to Fourier's approach in the study of partial differential equations, the Cauchy-Euler equation is significant in the theory of linear differential equations.

2Prove that mixed partial derivatives are equal:

It is given that for the differential equation

ay"+by'+cy=0

Where a, b, c are constants, the characteristic equation is,

ar²+br+c=0

For which there is a repeated root at r₀ and one solution of the equation is

y₁=e^r₀^t

Using operator approach, if r₀ is a repeated root, then

L[w] (e^t) = a (r-r₀)² e^rt

You observe that, the righthand side of the above equation has a factor (r-r₀)², so taking the partial derivative, w.r.t. r and setting r=r₀ you will get 0,

𝛿 /𝛿r [L [w] (t)] |_r=r₀= [a(r-r₀)² (re^rt)+2a (r-r₀) e^rt]|_r=r₀=0

You also note that w(r,e^t)=e^rt has continuous partial derivatives of all orders with respect to both and hence the mixed partial derivates are equal.

𝛿²w/𝛿r𝛿t² = 𝛿²w/𝛿r²𝛿t ,

𝛿²w/𝛿r𝛿t = 𝛿²w/𝛿r𝛿t

3Find the second solution:

Consequently, for a differential operator L, you have,

𝛿/𝛿r L[w] = 𝛿/𝛿r {a 𝛿²w/𝛿t² +b 𝛿w/𝛿t+cw}

=a 𝛿³w/𝛿r𝛿t² +b 𝛿²w/𝛿r𝛿t +c 𝛿w/𝛿r

=a 𝛿³w/𝛿t²𝛿r +b 𝛿²w/𝛿t𝛿r +c 𝛿w/𝛿r

=L [𝛿w/𝛿r]

From above, you can say that the operators 𝛿/𝛿r and L are commutative. Thus,

L [𝛿w/𝛿r]|_r=r₀=0

Therefore, in the case of repeated roots at r₀, the second linearly independent solution is,

y₂(x) = 𝛿w/𝛿r (r₀ , e^t)