2.5.12 problem 12

Internal problem ID [8649]
Book : Own collection of miscellaneous problems
Section : section 5.0
Problem number : 12
Date solved : Thursday, December 12, 2024 at 09:37:21 AM
CAS classification : [[_2nd_order, _missing_y]]

Solve

\begin{align*} y^{\prime \prime }+y^{\prime }&=\frac {1}{x} \end{align*}

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.

\[ y^{\prime \prime }+y^{\prime } = 0 \]

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

\begin{align*} y^{\prime \prime }+p(x) y^{\prime } + q(x) y &=0 \end{align*}

Where

\begin{align*} p(x) &= 1\\ q(x) &= 0\\ \end{align*}

Table 2.120: Table \(p(x),q(x)\) singularites.

\(p(x)=1\)
singularity	type

\(q(x)=0\)
singularity	type

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

Regular singular points : \([]\)

Irregular singular points : \([\infty ]\)

Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )+\frac {d}{d x}y \left (x \right )=\frac {1}{x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right ) \\ \bullet & {} & \textrm {Characteristic polynomial of homogeneous ODE}\hspace {3pt} \\ {} & {} & r^{2}+r =0 \\ \bullet & {} & \textrm {Factor the characteristic polynomial}\hspace {3pt} \\ {} & {} & r \left (r +1\right )=0 \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (-1, 0\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (x \right )={\mathrm e}^{-x} \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (x \right )=1 \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & y \left (x \right )=\mathit {C1} y_{1}\left (x \right )+\mathit {C2} y_{2}\left (x \right )+y_{p}\left (x \right ) \\ \bullet & {} & \textrm {Substitute in solutions of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y \left (x \right )=\mathit {C1} \,{\mathrm e}^{-x}+\mathit {C2} +y_{p}\left (x \right ) \\ \square & {} & \textrm {Find a particular solution}\hspace {3pt} y_{p}\left (x \right )\hspace {3pt}\textrm {of the ODE}\hspace {3pt} \\ {} & \circ & \textrm {Use variation of parameters to find}\hspace {3pt} y_{p}\hspace {3pt}\textrm {here}\hspace {3pt} f \left (x \right )\hspace {3pt}\textrm {is the forcing function}\hspace {3pt} \\ {} & {} & \left [y_{p}\left (x \right )=-y_{1}\left (x \right ) \left (\int \frac {y_{2}\left (x \right ) f \left (x \right )}{W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )}d x \right )+y_{2}\left (x \right ) \left (\int \frac {y_{1}\left (x \right ) f \left (x \right )}{W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )}d x \right ), f \left (x \right )=\frac {1}{x}\right ] \\ {} & \circ & \textrm {Wronskian of solutions of the homogeneous equation}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )=\left [\begin {array}{cc} {\mathrm e}^{-x} & 1 \\ -{\mathrm e}^{-x} & 0 \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )={\mathrm e}^{-x} \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} y_{p}\left (x \right ) \\ {} & {} & y_{p}\left (x \right )=-{\mathrm e}^{-x} \left (\int \frac {{\mathrm e}^{x}}{x}d x \right )+\int \frac {1}{x}d x \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (x \right )={\mathrm e}^{-x} \mathrm {Ei}_{1}\left (-x \right )+\ln \left (x \right ) \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y \left (x \right )=\mathit {C1} \,{\mathrm e}^{-x}+\mathit {C2} +{\mathrm e}^{-x} \mathrm {Ei}_{1}\left (-x \right )+\ln \left (x \right ) \end {array} \]

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
-> Calling odsolve with the ODE`, diff(_b(_a), _a) = -(_b(_a)*_a-1)/_a, _b(_a)`   *** Sublevel 2 *** 
   Methods for first order ODEs: 
   --- Trying classification methods --- 
   trying a quadrature 
   trying 1st order linear 
   <- 1st order linear successful 
<- high order exact linear fully integrable successful`
 

Maple dsolve solution

Solving time : 0.042 (sec)
Leaf size : maple_leaf_size

dsolve(diff(y(x),x)+diff(diff(y(x),x),x) = 1/x,y(x), 
       series,x=0)
 

\[ \text {No solution found} \]

Mathematica DSolve solution

Solving time : 0.024 (sec)
Leaf size : 159

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

\[ y(x)\to -\frac {x^6}{4320}-\frac {x^5}{600}-\frac {x^4}{96}-\frac {x^3}{18}-\frac {x^2}{4}+c_2 \left (-\frac {x^5}{720}+\frac {x^4}{120}-\frac {x^3}{24}+\frac {x^2}{6}-\frac {x}{2}+1\right ) x+\left (-\frac {x^5}{720}+\frac {x^4}{120}-\frac {x^3}{24}+\frac {x^2}{6}-\frac {x}{2}+1\right ) x \left (\frac {x^6}{2160}+\frac {x^5}{600}+\frac {x^4}{96}+\frac {x^3}{18}+\frac {x^2}{4}+x+\log (x)\right )-x+c_1 \]