SOLUTION: Reduce the given cauchy-Euler equation to a linear differential equation with constant coefficient. X^2.d^2y/dx^2 -2Xdy/dx+2y=X^3. ? (8 marks)

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Question 1025056: Reduce the given cauchy-Euler equation to a linear differential equation with constant coefficient.
X^2.d^2y/dx^2 -2Xdy/dx+2y=X^3. ? (8 marks)
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
X%5E2d%5E2y%2Fdx%5E2+-2Xdy%2Fdx%2B2y=x%5E3
Use the transformation x+=+e%5Ez
==> dx%2Fdz+=+e%5Ez ==> dy%2Fdz+=+%28dy%2Fdx%29%2A%28dx%2Fdz%29+=+e%5Ez%2A%28dy%2Fdx%29+=+x%28dy%2Fdx%29
==>
==>
==> d%5E2y%2Fdz%5E2+-+e%5Ez%2A%28dy%2Fdx%29+-+2e%5Ez%2A%28dy%2Fdx%29+%2B+2y+=+x%5E3
==>
==> d%5E2y%2Fdz%5E2+-+3dy%2Fdz+%2B+2y+=+e%5E%283z%29 <--Equation (A)
Now try the particular solution y%5Bp%5D+=+Ce%5E%283z%29
By substitution into Equation (A), we get C = 1/2.
==> the general solution to Equation (A) is
y+=+Ae%5Ez+%2B+Be%5E%282z%29+%2B+e%5E%283z%29%2F2 <-- Equation (B)
==> Now from x+=+e%5Ez and by direct substitution into Equation (B), we get
highlight%28y+=+Ax+%2B+Bx%5E2+%2B+x%5E3%2F2%29+,
the general solution to the original Cauchy-Euler equation.