Linear Differential Equations

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Legendre equation

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Euler's equation

Euler's equation states (z^2)w'' + αzw' + βw = 0 where w'=dw/dz If we substitute in w=z^u into Euler's equation, we find the so-called indicial equation: u^2 + (α-1)u +β=0 Use the change of variable t=ln(z) in the Euler equation to show that the Euler equation has the linearly independent pair of solutions z^u1 and z^u2 if the indicial equation has roots u1 and u2 or z^u1, (z^u1)ln(z) if there is a single root of multiplicity two.

General Solution to the Differential Equation

Help! I am having a problem with this homogenous differential equation (y^2 + yx)dx + x(^2)dy=0 I tried substitution of x = vy, dx = vdy + ydv, but I am not quite getting it right. Please help!

Difficult Diff. EQ

This problem is really giving me issues, as I do not even know how to set it up to solve it. Please solve this problem in as much detail as possible. I guess the number 1 in the parentheses is what is confusing me. Thank you so much! (1 + x^2 + Y^2 +(x^2)(y^2))dy = (y^2)dx