2.5.14 Problem 14

Internal problem ID [8975]
Book : Own collection of miscellaneous problems
Section : section 5.0
Problem number : 14
Date solved : Sunday, March 30, 2025 at 01:57:40 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

Solve

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

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.

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

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

$$\begin{array}{rl}p(x)& =1\\ q(x)& =1\end{array}$$

Table 2.124: Table $p(x),q(x)$ singularites.

$p(x)=1$
singularity	type

$q(x)=1$
singularity	type

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

Regular singular points : $[]$

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

Since $x=0$ is not an ordinary point, then we will now check if it is a regular singular point. Unable to solve since $x=0$ is not regular singular point. Terminating.

✗ Maple

Order:=6; 
ode:=diff(diff(y(x),x),x)+diff(y(x),x)+y(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 
trying high order exact linear fully integrable 
trying differential order: 2; linear nonhomogeneous with symmetry [0,1] 
trying a double symmetry of the form [xi=0, eta=F(x)] 
-> Try solving first the homogeneous part of the ODE 
   checking if the LODE has constant coefficients 
   <- constant coefficients successful 
<- solving first the homogeneous part of the ODE successful
 

Maple step by step

$$\begin{array}{lll}& & \text{Let’s solve}\\ & & \frac{d}{dx}\frac{d}{dx}y\left(x\right)+\frac{d}{dx}y\left(x\right)+y\left(x\right)=\frac{1}{x}\\ \bullet & & \text{Highest derivative means the order of the ODE is}2\\ & & \frac{d}{dx}\frac{d}{dx}y\left(x\right)\\ \bullet & & \text{Characteristic polynomial of homogeneous ODE}\\ & & r^2+r+1=0\\ \bullet & & \text{Use quadratic formula to solve for}r\\ & & r=\frac{(-1)\pm \left(\sqrt{-3}\right)}{2}\\ \bullet & & \text{Roots of the characteristic polynomial}\\ & & r=(-\frac{1}{2}-\frac{\mathrm{I}\sqrt{3}}{2},-\frac{1}{2}+\frac{\mathrm{I}\sqrt{3}}{2})\\ \bullet & & \text{1st solution of the homogeneous ODE}\\ & & y_1\left(x\right)={\mathrm{e}}^{-\frac{x}{2}}\cos \left(\frac{\sqrt{3}x}{2}\right)\\ \bullet & & \text{2nd solution of the homogeneous ODE}\\ & & y_2\left(x\right)={\mathrm{e}}^{-\frac{x}{2}}\sin \left(\frac{\sqrt{3}x}{2}\right)\\ \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}{\mathrm{e}}^{-\frac{x}{2}}\cos \left(\frac{\sqrt{3}x}{2}\right)+\mathit{C}\mathit{2}{\mathrm{e}}^{-\frac{x}{2}}\sin \left(\frac{\sqrt{3}x}{2}\right)+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}{\mathrm{e}}^{-\frac{x}{2}}\cos \left(\frac{\sqrt{3}x}{2}\right)& {\mathrm{e}}^{-\frac{x}{2}}\sin \left(\frac{\sqrt{3}x}{2}\right)\\ -\frac{{\mathrm{e}}^{-\frac{x}{2}}\cos \left(\frac{\sqrt{3}x}{2}\right)}{2}-\frac{{\mathrm{e}}^{-\frac{x}{2}}\sqrt{3}\sin \left(\frac{\sqrt{3}x}{2}\right)}{2}& -\frac{{\mathrm{e}}^{-\frac{x}{2}}\sin \left(\frac{\sqrt{3}x}{2}\right)}{2}+\frac{{\mathrm{e}}^{-\frac{x}{2}}\sqrt{3}\cos \left(\frac{\sqrt{3}x}{2}\right)}{2}\end{array}\right]\\ & \circ & \text{Compute Wronskian}\\ & & W(y_1\left(x\right),y_2\left(x\right))=\frac{\sqrt{3}{\mathrm{e}}^{-x}}{2}\\ & \circ & \text{Substitute functions into equation for}{y}_p\left(x\right)\\ & & y_p\left(x\right)=-\frac{2\sqrt{3}{\mathrm{e}}^{-\frac{x}{2}}(\int \frac{{\mathrm{e}}^{\frac{x}{2}}\sin \left(\frac{\sqrt{3}x}{2}\right)}{x}dx\cos \left(\frac{\sqrt{3}x}{2}\right)-\int \frac{{\mathrm{e}}^{\frac{x}{2}}\cos \left(\frac{\sqrt{3}x}{2}\right)}{x}dx\sin \left(\frac{\sqrt{3}x}{2}\right))}{3}\\ & \circ & \text{Compute integrals}\\ & & y_p\left(x\right)=-\frac{{\mathrm{e}}^{-\frac{x}{2}}((\mathrm{I}\cos \left(\frac{\sqrt{3}x}{2}\right)+\sin \left(\frac{\sqrt{3}x}{2}\right)){\mathrm{Ei}}_1(-\frac{x(1+\mathrm{I}\sqrt{3})}{2})-{\mathrm{Ei}}_1\left(\frac{x(\mathrm{I}\sqrt{3}-1)}{2}\right)(\mathrm{I}\cos \left(\frac{\sqrt{3}x}{2}\right)-\sin \left(\frac{\sqrt{3}x}{2}\right)))\sqrt{3}}{3}\\ \bullet & & \text{Substitute particular solution into general solution to ODE}\\ & & y\left(x\right)=\mathit{C}\mathit{1}{\mathrm{e}}^{-\frac{x}{2}}\cos \left(\frac{\sqrt{3}x}{2}\right)+\mathit{C}\mathit{2}{\mathrm{e}}^{-\frac{x}{2}}\sin \left(\frac{\sqrt{3}x}{2}\right)-\frac{{\mathrm{e}}^{-\frac{x}{2}}((\mathrm{I}\cos \left(\frac{\sqrt{3}x}{2}\right)+\sin \left(\frac{\sqrt{3}x}{2}\right)){\mathrm{Ei}}_1(-\frac{x(1+\mathrm{I}\sqrt{3})}{2})-{\mathrm{Ei}}_1\left(\frac{x(\mathrm{I}\sqrt{3}-1)}{2}\right)(\mathrm{I}\cos \left(\frac{\sqrt{3}x}{2}\right)-\sin \left(\frac{\sqrt{3}x}{2}\right)))\sqrt{3}}{3}\end{array}$$

✓ Mathematica. Time used: 0.06 (sec). Leaf size: 152

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

$$y(x)\to c_{2}x(-\frac{x^4}{120}+\frac{x^3}{24}-\frac{x}{2}+1)+c_1(\frac{x^3}{6}-\frac{x^2}{2}+1)+x(-\frac{x^4}{120}+\frac{x^3}{24}-\frac{x}{2}+1)(\frac{41x^6}{4320}+\frac{x^5}{120}-\frac{x^4}{96}-\frac{x^3}{18}+x+\log (x))+(\frac{x^3}{6}-\frac{x^2}{2}+1)(-\frac{x^6}{180}+\frac{x^5}{600}+\frac{x^4}{96}-\frac{x^2}{4}-x)$$

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) + Derivative(y(x), x) + 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 y(x) + Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 1/x does not match hint 2nd_power_series_regular