Consider the differential equation: xy + (3x-1)y' + y = 0 for x > 0. (14a)...

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Consider the differential equation: x²y" + (3x-1)y' + y = 0 for x > 0. (14a) Show that x = 0 is an irregular singular point of the above differential equation. (14b) Let us, nevertheless, attempt a solution of the form: aoxm + a₁xm+1 + a₂xm+2+ a3xm+3+... with ao 0. Plugging the above expansion for the unknown in the above differential equation, show that the exponent m must be zero. (14c) Determine the coefficients a1, 42,... in terms of ag. (144) Does the series solution, thus built, have a positive radius of convergence? Consider the differential equation: x²y" + (3x-1)y' + y = 0 for x > 0. (14a) Show that x = 0 is an irregular singular point of the above differential equation. (14b) Let us, nevertheless, attempt a solution of the form: aoxm + a₁xm+1 + a₂xm+2+ a3xm+3+... with ao 0. Plugging the above expansion for the unknown in the above differential equation, show that the exponent m must be zero. (14c) Determine the coefficients a1, 42,... in terms of ag. (144) Does the series solution, thus built, have a positive radius of convergence?

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