91Ó°ÊÓ

\(y^{\prime \prime}-y^{\prime}=0, y(0)=3, y^{\prime}(0)=2\)

The differential equation \(y^{\prime \prime}+p(x) y^{\prime}+q(x) y\) \(=0\) has a singular point at infinity if after substitution of \(w=1 / x\) the resulting equation has a singular point at \(w=0\). Similarly, the equation has an ordinary point at infinity if the transformed equation has an ordinary point at \(w=0\). Use the chain rule and the substitution \(w=1 / x\) to show that the differential equation \(y^{\prime \prime}+p(x) y^{\prime}+q(x) y=\) 0 , where ' is with respect to \(x\), is equivalent to $$ \frac{d^{2} y}{d w^{2}}+\left(\frac{2}{w}+\frac{p(1 / w)}{w^{2}}\right) \frac{d y}{d w}+\frac{q(1 / w)}{w^{4}} y=0 . $$

\(y^{\prime \prime}+y^{\prime}+x y=\cos x, y(0)=0, y^{\prime}(0)=1\)

The hypergeometric equation is given by \(x(1-x) y^{\prime \prime}+[c-(a+b+1) x] y^{\prime}-a b y=0\), where \(a, b\), and \(c\) are constants. (a) Show that \(x=0\) and \(x=1\) are regular singular points. (b) Show that the roots of the indicial equation for the series \(\sum_{n=0}^{\infty} a_{n} x^{n+r}\) are \(r=0\) and \(r=1-c\). (c) Show that for \(r=0\), the solution obtained with the Method of Frobenius is $$ \begin{aligned} y_{1}=& 1+\frac{a b}{1 ! c} x+\frac{a(a+1) b(b+1)}{2 ! c(c+1)} x^{2} \\ &+\frac{a(a+1)(a+2) b(b+1)(b+2)}{3 ! c(c+1)(c+2)} x^{3} \\ &+\cdots \end{aligned} $$ where \(c \neq 0,-1,-2, \ldots\). This series is called the hypergeometric series. Its sum, denoted \(F(a, b, c ; x)\), is called the hypergeometric function. (d) Show that \(F(1, b, b ; x)\) \(=1 /(1-x)\). (e) Find the solution of the equation that corresponds to \(r=1-c\).

\(y^{\prime \prime}+100 y=0, y(0)=1, y^{\prime}(0)=10\)

\(x^{2} y^{\prime \prime}+x y^{\prime}+y=x^{-2}\)

\(y^{(4)}-16 y=0\)

\(x^{3} y^{\prime \prime \prime}+15 x^{2} y^{\prime \prime}+54 x y^{\prime}+42 y=0, y(1)=5\), \(y^{\prime}(1)=0, y^{\prime \prime}(1)=0\)

\(y^{\prime \prime \prime}-y=0, y(0)=0, y^{\prime}(0)=0, y^{\prime \prime}(0)=3\)

Can the Wronskian be zero at only one value of \(t\) on \(I\) ? Hint: Use Abel's formula.