
# 7.4E: Regular Singular Points Euler Equations (Exercises)

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In Exercises [exer:7.4.1} –[exer:7.4.18} find the general solution of the given Euler equation on $$(0,\infty)$$.

[exer:7.4.1] $$x^2y''+7xy'+8y=0$$

[exer:7.4.2] $$x^2y''-7xy'+7y=0$$

[exer:7.4.3] $$x^2y''-xy'+y=0$$

[exer:7.4.4] $$x^2y''+5xy'+4y=0$$

[exer:7.4.5] $$x^2y''+xy'+y=0$$

[exer:7.4.6] $$x^2y''-3xy'+13y=0$$

[exer:7.4.7] $$x^2y''+3xy'-3y=0$$

[exer:7.4.8] $$12x^2y''-5xy''+6y=0$$

[exer:7.4.9] $$4x^2y''+8xy'+y=0$$

[exer:7.4.10] $$3x^2y''-xy'+y=0$$

[exer:7.4.11] $$2x^2y''-3xy'+2y=0$$

[exer:7.4.12] $$x^2y''+3xy'+5y=0$$

[exer:7.4.13] $$9x^2y''+15xy'+y=0$$

[exer:7.4.14] $$x^2y''-xy'+10y=0$$

[exer:7.4.15] $$x^2y''-6y=0$$

[exer:7.4.16] $$2x^2y''+3xy'-y=0$$

[exer:7.4.17] $$x^2y''-3xy'+4y=0$$

[exer:7.4.18] $$2x^2y''+10xy'+9y=0$$

[exer:7.4.19]

Adapt the proof of Theorem [thmtype:7.4.3} to show that $$y=y(x)$$ satisfies the Euler equation

$\label{eq:7.4.{exer:7.4.19}A} ax^2y''+bxy'+cy=0\nonumber$

on $$(-\infty,0)$$ if and only if $$Y(t)=y(-e^t)$$

$a {d^2Y\over dt^2}+(b-a){dY\over dt}+cY=0.\nonumber$

on $$(-\infty,\infty)$$.

Use (a) to show that the general solution of Equation \ref{eq:7.4.{exer:7.4.19}A} on $$(-\infty,0)$$ is

\begin{aligned} y&=&c_1|x|^{r_1}+c_2|x|^{r_2}\mbox{ if r_1 and r_2 are distinct real numbers; } \\ y&=&|x|^{r_1}(c_1+c_2\ln|x|)\mbox{ if r_1=r_2; } \\ y&=&|x|^{\lambda}\left[c_1\cos\left(\omega\ln|x|\right)+ c_2\sin\left(\omega\ln|x| \right)\right]\mbox{ if r_1,r_2=\lambda\pm i\omega with \omega>0}.\end{aligned}\nonumber

[exer:7.4.20] Use reduction of order to show that if

$ar(r-1)+br+c=0\nonumber$

has a repeated root $$r_1$$ then $$y=x^{r_1}(c_1+c_2\ln x)$$ is the general solution of

$ax^2y''+bxy'+cy=0\nonumber$

on $$(0,\infty)$$.

[exer:7.4.21] A nontrivial solution of

$P_0(x)y''+P_1(x)y'+P_2(x)y=0\nonumber$

is said to be oscillatory on an interval $$(a,b)$$ if it has infinitely many zeros on $$(a,b)$$. Otherwise $$y$$ is said to be nonoscillatory on $$(a,b)$$. Show that the equation

$x^2y''+ky=0 \quad (k=\; \mbox{constant})\nonumber$

has oscillatory solutions on $$(0,\infty)$$ if and only if $$k>1/4$$.

[exer:7.4.22] In Example

Example $$\PageIndex{1}$$:

Add text here. For the automatic number to work, you need to add the “AutoNum” template (preferably at7.4.2} we saw that $$x_0=1$$ and $$x_0=-1$$ are regular singular points of Legendre’s equation

$(1-x^2)y''-2xy'+\alpha(\alpha+1)y=0. \tag{A}$

Introduce the new variables $$t=x-1$$ and $$Y(t)=y(t+1)$$, and show that $$y$$ is a solution of (A) if and only if $$Y$$ is a solution of

$t(2+t){d^2Y\over dt^2}+2(1+t){dY\over dt}-\alpha(\alpha+1)Y=0,\nonumber$

which has a regular singular point at $$t_0=0$$.

Introduce the new variables $$t=x+1$$ and $$Y(t)=y(t-1)$$, and show that $$y$$ is a solution of (A) if and only if $$Y$$ is a solution of

$t(2-t){d^2Y\over dt^2}+2(1-t){dY\over dt}+\alpha(\alpha+1)Y=0,\nonumber$

which has a regular singular point at $$t_0=0$$.

[exer:7.4.23] Let $$P_0,P_1$$, and $$P_2$$ be polynomials with no common factor, and suppose $$x_0\ne0$$ is a singular point of

$P_0(x)y''+P_1(x)y'+P_2(x)y=0. \tag{A}$

Let $$t=x-x_0$$ and $$Y(t)=y(t+x_0)$$.

Show that $$y$$ is a solution of (A) if and only if $$Y$$ is a solution of

$R_0(t){d^2Y\over dt^2}+R_1(t){dY\over dt}+R_2(t)Y=0. \tag{B}] where \[R_i(t)=P_i(t+x_0),\quad i=0,1,2.\nonumber$

Show that $$R_0$$, $$R_1$$, and $$R_2$$ are polynomials in $$t$$ with no common factors, and $$R_0(0)=0$$; thus, $$t_0=0$$ is a singular point of (B).