Internal problem ID [4862]
Internal file name [OUTPUT/4355_Sunday_June_05_2022_01_05_49_PM_32176453/index.tex]
Book: Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley.
2006
Section: Chapter 8, Ordinary differential equations. Section 7. Other second-Order equations.
page 435
Problem number: 29.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "reduction_of_order", "exact linear second order ode", "second_order_integrable_as_is", "second_order_ode_non_constant_coeff_transformation_on_B"
Maple gives the following as the ode type
[[_2nd_order, _exact, _linear, _homogeneous]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (x +1\right ) y^{\prime }-y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x +1 \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 \[ p \left (x \right ) = \frac {x +1}{x^{2}} \] Therefore \begin{align*} y_{2}\left (x \right ) &= \left (x +1\right ) \left (\int \frac {{\mathrm e}^{-\left (\int \frac {x +1}{x^{2}}d x \right )}}{\left (x +1\right )^{2}}d x \right ) \\ y_{2}\left (x \right ) &= x +1 \int \frac {{\mathrm e}^{\frac {1}{x}-\ln \left (x \right )}}{\left (x +1\right )^{2}} , dx \\ y_{2}\left (x \right ) &= \left (x +1\right ) \left (\int \frac {{\mathrm e}^{\frac {1}{x}}}{x \left (x +1\right )^{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 ) \\ &= \left (x +1\right ) c_{1} -c_{2} x \,{\mathrm e}^{\frac {1}{x}} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \left (x +1\right ) c_{1} -c_{2} x \,{\mathrm e}^{\frac {1}{x}} \\ \end{align*}
Verification of solutions
\[ y = \left (x +1\right ) c_{1} -c_{2} x \,{\mathrm e}^{\frac {1}{x}} \] 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^2*diff(y(x),x$2)+(x+1)*diff(y(x),x)-y(x)=0,x+1],singsol=all)
\[ y \left (x \right ) = c_{2} {\mathrm e}^{\frac {1}{x}} x +c_{1} x +c_{1} \]
✓ Solution by Mathematica
Time used: 0.077 (sec). Leaf size: 21
DSolve[x^2*y''[x]+(x+1)*y'[x]-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 e^{\frac {1}{x}} x+c_2 (x+1) \]