Internal problem ID [12195]
Internal file name [OUTPUT/10848_Thursday_September_21_2023_05_48_01_AM_14234251/index.tex]
Book: Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS,
MOSCOW, Third printing 1977.
Section: Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER.
Problems page 172
Problem number: Problem 47.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "reduction_of_order", "second_order_change_of_variable_on_y_method_2", "second_order_ode_non_constant_coeff_transformation_on_B"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\[ \boxed {x^{3} y^{\prime \prime }-y^{\prime } x +y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*}
Given one basis solution \(y_{1}\left (x \right )\), then the second basis solution is given by \[ y_{2}\left (x \right ) = y_{1} \left (\int \frac {{\mathrm e}^{-\left (\int p d x \right )}}{y_{1}^{2}}d x \right ) \] Where \(p(x)\) is the coefficient of \(y^{\prime }\) when the ode is written in the normal form \[ y^{\prime \prime }+p \left (x \right ) y^{\prime }+q \left (x \right ) y = f \left (x \right ) \] Looking at the ode to solve shows that \[ p \left (x \right ) = -\frac {1}{x^{2}} \] Therefore \begin{align*} y_{2}\left (x \right ) &= x \left (\int \frac {{\mathrm e}^{-\left (\int -\frac {1}{x^{2}}d x \right )}}{x^{2}}d x \right ) \\ y_{2}\left (x \right ) &= x \int \frac {{\mathrm e}^{-\frac {1}{x}}}{x^{2}} , dx \\ y_{2}\left (x \right ) &= x \left (\int \frac {{\mathrm e}^{-\frac {1}{x}}}{x^{2}}d x \right ) \\ y_{2}\left (x \right ) &= x \,{\mathrm e}^{-\frac {1}{x}} \\ \end{align*} Hence the solution is \begin{align*} y &= c_{1} y_{1}\left (x \right )+c_{2} y_{2}\left (x \right ) \\ &= c_{1} x +c_{2} x \,{\mathrm e}^{-\frac {1}{x}} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} x +c_{2} x \,{\mathrm e}^{-\frac {1}{x}} \\ \end{align*}
Verification of solutions
\[ y = c_{1} x +c_{2} x \,{\mathrm e}^{-\frac {1}{x}} \] Verified OK.
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] <- linear_1 successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 16
dsolve([x^3*diff(y(x),x$2)-x*diff(y(x),x)+y(x)=0,x],singsol=all)
\[ y \left (x \right ) = \left ({\mathrm e}^{-\frac {1}{x}} c_{1} +c_{2} \right ) x \]
✓ Solution by Mathematica
Time used: 0.095 (sec). Leaf size: 20
DSolve[x^3*y''[x]-x*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x \left (c_2 e^{-1/x}+c_1\right ) \]