(x^3D^3+3x^2D^2+xD+1)y=xlogx #CauchyEulerLinearEquation L955

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Hello, People!

Here is a video of solving a differential equation which is of Cauchy's form. By reducing the given Cauchy's equation(linear differential equation with variable coefficient) to linear differential equation with constant coefficient.
Watch with patience till end, or skip some part.

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Chinnaiah Kalpana🌞

Note:

* Cauchy's Homogeneous Linear Differential Equations:

An equation of the form

a0 x^n D^n y +a1 x^n-1 D^n-1 y + ....+ an-1 x D y +an y =Q(x)

where a0, a1,...,an are constants with a0 non zero, is called Cauchy's homogeneous libear differential equation (or) Euler-Cauchy linear equation (or) Euler's linear equation. This can be reduced to a linear differential equation with constant coefficients.
Put x = e^z then z = logx

dydx =dy/dz dz/dx =dy/dz.1/x
then x dy/dx = dy/dz
then xD y = θ y
then xD=θ

X^2D^2 = θ(θ-1)
x^3D^3=θ(θ-1)(θ-2)
....

Later use known methods to find general solution of the given problem.

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