Paper 2, Section II, B

Differential Equations | Part IA, 2018

Consider the differential equation

x2d2ydx2+xdydx(x2+α2)y=0x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}-\left(x^{2}+\alpha^{2}\right) y=0

What values of xx are ordinary points of the differential equation? What values of xx are singular points of the differential equation, and are they regular singular points or irregular singular points? Give clear definitions of these terms to support your answers.

For α\alpha not equal to an integer there are two linearly independent power series solutions about x=0x=0. Give the forms of the two power series and the recurrence relations that specify the relation between successive coefficients. Give explicitly the first three terms in each power series.

For α\alpha equal to an integer explain carefully why the forms you have specified do not give two linearly independent power series solutions. Show that for such values of α\alpha there is (up to multiplication by a constant) one power series solution, and give the recurrence relation between coefficients. Give explicitly the first three terms.

If y1(x)y_{1}(x) is a solution of the above second-order differential equation then

y2(x)=y1(x)cx1s[y1(s)]2dsy_{2}(x)=y_{1}(x) \int_{c}^{x} \frac{1}{s\left[y_{1}(s)\right]^{2}} d s

where cc is an arbitrarily chosen constant, is a second solution that is linearly independent of y1(x)y_{1}(x). For the case α=1\alpha=1, taking y1(x)y_{1}(x) to be a power series, explain why the second solution y2(x)y_{2}(x) is not a power series.

[You may assume that any power series you use are convergent.]

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