2.5.11 Problem 11

Internal problem ID [8972]
Book : Own collection of miscellaneous problems
Section : section 5.0
Problem number : 11
Date solved : Wednesday, March 05, 2025 at 07:13:27 AM
CAS classification : [[_2nd_order, _quadrature]]

Solve

$$\begin{array}{rl}{y}^{\prime \prime }& =\frac{1}{x}\end{array}$$

Using series expansion around $x=0$

The type of the expansion point is first determined. This is done on the homogeneous part of the ODE.

$$y^{\prime \prime }=0$$

The following is summary of singularities for the above ode. Writing the ode as

$$\begin{array}{rl}{y}^{\prime \prime }+p(x)y^{\prime }+q(x)y& =0\end{array}$$

Where

$$\begin{array}{rl}p(x)& =0\\ q(x)& =0\end{array}$$

Table 2.117: Table $p(x),q(x)$ singularites.

$p(x)=0$
singularity	type

$q(x)=0$
singularity	type

Combining everything together gives the following summary of singularities for the ode as

Regular singular points : $[\mathrm{\infty }]$

Irregular singular points : $[]$

✗ Maple

Order:=6; 
ode:=diff(diff(y(x),x),x) = 1/x; 
dsolve(ode,y(x),type='series',x=0);
 

$$\text{No solution found}$$

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
<- quadrature successful`
 

Maple step by step

$$\begin{array}{lll}& & \text{Let’s solve}\\ & & \frac{d^2}{d{x}^2}y\left(x\right)=\frac{1}{x}\\ \bullet & & \text{Highest derivative means the order of the ODE is}2\\ & & \frac{d^2}{d{x}^2}y\left(x\right)\\ \bullet & & \text{Characteristic polynomial of homogeneous ODE}\\ & & r^2=0\\ \bullet & & \text{Use quadratic formula to solve for}r\\ & & r=\frac{0\pm \left(\sqrt{0}\right)}{2}\\ \bullet & & \text{Roots of the characteristic polynomial}\\ & & r=0\\ \bullet & & \text{1st solution of the homogeneous ODE}\\ & & y_1\left(x\right)=1\\ \bullet & & \text{Repeated root, multiply}{y}_1\left(x\right)\text{by}x\text{to ensure linear independence}\\ & & y_2\left(x\right)=x\\ \bullet & & \text{General solution of the ODE}\\ & & y\left(x\right)=\mathit{C}\mathit{1}{y}_1\left(x\right)+\mathit{C}\mathit{2}{y}_2\left(x\right)+y_p\left(x\right)\\ \bullet & & \text{Substitute in solutions of the homogeneous ODE}\\ & & y\left(x\right)=\mathit{C}\mathit{1}+\mathit{C}\mathit{2}x+y_p\left(x\right)\\ ◻& & \text{Find a particular solution}{y}_p\left(x\right)\text{of the ODE}\\ & \circ & \text{Use variation of parameters to find}{y}_p\text{here}f\left(x\right)\text{is the forcing function}\\ & & [y_p\left(x\right)=-y_1\left(x\right)(\int \frac{y_2\left(x\right)f\left(x\right)}{W(y_1\left(x\right),y_2\left(x\right))}dx)+y_2\left(x\right)(\int \frac{y_1\left(x\right)f\left(x\right)}{W(y_1\left(x\right),y_2\left(x\right))}dx),f\left(x\right)=\frac{1}{x}]\\ & \circ & \text{Wronskian of solutions of the homogeneous equation}\\ & & W(y_1\left(x\right),y_2\left(x\right))=\left[\begin{array}{cc}1& x\\ 0& 1\end{array}\right]\\ & \circ & \text{Compute Wronskian}\\ & & W(y_1\left(x\right),y_2\left(x\right))=1\\ & \circ & \text{Substitute functions into equation for}{y}_p\left(x\right)\\ & & y_p\left(x\right)=-(\int 1dx)+(\int \frac{1}{x}dx)x\\ & \circ & \text{Compute integrals}\\ & & y_p\left(x\right)=x(-1+\ln \left(x\right))\\ \bullet & & \text{Substitute particular solution into general solution to ODE}\\ & & y\left(x\right)=\mathit{C}\mathit{1}+\mathit{C}\mathit{2}x+x(-1+\ln \left(x\right))\end{array}$$

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 17

ode=D[y[x],{x,2}]==1/x; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

$$y(x)\to -x+x\log (x)+c_{2}x+c_1$$

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) - 1/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE Derivative(y(x), (x, 2)) - 1/x does not match hint 2nd_power_series_regular