Internal problem ID [7307]
Internal file name [OUTPUT/6293_Sunday_June_05_2022_04_37_28_PM_25391719/index.tex]
Book: Own collection of miscellaneous problems
Section: section 5.0
Problem number: 14.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_linear_constant_coeff"
Maple gives the following as the ode type
[[_2nd_order, _linear, _nonhomogeneous]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+y^{\prime }+y=\frac {1}{x}} \] With the expansion point for the power series method at \(x = 0\).
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 {align*} y^{\prime \prime }+p(x) y^{\prime } + q(x) y &=0 \end {align*}
Where \begin {align*} p(x) &= 1\\ q(x) &= 1\\ \end {align*}
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([]\)
Irregular singular points : \([\infty ]\)
Verification of solutions N/A
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime }+y^{\prime }+y=\frac {1}{x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Characteristic polynomial of homogeneous ODE}\hspace {3pt} \\ {} & {} & r^{2}+r +1=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {\left (-1\right )\pm \left (\sqrt {-3}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}, -\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (x \right )={\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (x \right )={\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & y=c_{1} y_{1}\left (x \right )+c_{2} y_{2}\left (x \right )+y_{p}\left (x \right ) \\ \bullet & {} & \textrm {Substitute in solutions of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y=c_{1} {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+c_{2} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) {\mathrm e}^{-\frac {x}{2}}+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}^{-\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 & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )=\frac {\sqrt {3}\, {\mathrm e}^{-x}}{2} \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} y_{p}\left (x \right ) \\ {} & {} & y_{p}\left (x \right )=-\frac {2 \,{\mathrm e}^{-\frac {x}{2}} \sqrt {3}\, \left (\cos \left (\frac {\sqrt {3}\, x}{2}\right ) \left (\int \frac {{\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{x}d x \right )-\sin \left (\frac {\sqrt {3}\, x}{2}\right ) \left (\int \frac {{\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{x}d x \right )\right )}{3} \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (x \right )=-\frac {{\mathrm e}^{-\frac {x}{2}} \sqrt {3}\, \left (\left (\mathrm {I} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\sin \left (\frac {\sqrt {3}\, x}{2}\right )\right ) \mathrm {Ei}_{1}\left (-\frac {x \left (1+\mathrm {I} \sqrt {3}\right )}{2}\right )-\mathrm {Ei}_{1}\left (\frac {x \left (\mathrm {I} \sqrt {3}-1\right )}{2}\right ) \left (\mathrm {I} \cos \left (\frac {\sqrt {3}\, x}{2}\right )-\sin \left (\frac {\sqrt {3}\, x}{2}\right )\right )\right )}{3} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y=c_{1} {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+c_{2} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) {\mathrm e}^{-\frac {x}{2}}-\frac {{\mathrm e}^{-\frac {x}{2}} \sqrt {3}\, \left (\left (\mathrm {I} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\sin \left (\frac {\sqrt {3}\, x}{2}\right )\right ) \mathrm {Ei}_{1}\left (-\frac {x \left (1+\mathrm {I} \sqrt {3}\right )}{2}\right )-\mathrm {Ei}_{1}\left (\frac {x \left (\mathrm {I} \sqrt {3}-1\right )}{2}\right ) \left (\mathrm {I} \cos \left (\frac {\sqrt {3}\, x}{2}\right )-\sin \left (\frac {\sqrt {3}\, x}{2}\right )\right )\right )}{3} \end {array} \]
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`
✗ Solution by Maple
Order:=6; dsolve(diff(y(x),x$2)+diff(y(x),x)+y(x)=1/x,y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.055 (sec). Leaf size: 152
AsymptoticDSolveValue[y''[x]+y'[x]+y[x]==1/x,y[x],{x,0,5}]
\[ y(x)\to c_2 x \left (-\frac {x^4}{120}+\frac {x^3}{24}-\frac {x}{2}+1\right )+c_1 \left (\frac {x^3}{6}-\frac {x^2}{2}+1\right )+x \left (-\frac {x^4}{120}+\frac {x^3}{24}-\frac {x}{2}+1\right ) \left (\frac {41 x^6}{4320}+\frac {x^5}{120}-\frac {x^4}{96}-\frac {x^3}{18}+x+\log (x)\right )+\left (\frac {x^3}{6}-\frac {x^2}{2}+1\right ) \left (-\frac {x^6}{180}+\frac {x^5}{600}+\frac {x^4}{96}-\frac {x^2}{4}-x\right ) \]