Internal problem ID [11320]
Internal file name [OUTPUT/10306_Tuesday_December_27_2022_04_06_09_AM_65226813/index.tex]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IX, Miscellaneous methods for solving equations of higher order than first.
Article 59. Linear equations with particular integral known. Page 136
Problem number: Ex 1.
ODE order: 3.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_3rd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {\left (x^{2}-2 x +2\right ) y^{\prime \prime \prime }-x^{2} y^{\prime \prime }+2 x y^{\prime }-2 y=0} \] Unable to solve this ODE.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying high order exact linear fully integrable trying to convert to a linear ODE with constant coefficients trying differential order: 3; missing the dependent variable Equation is the LCLM of -1/x*y(x)+diff(y(x),x), -2/x*y(x)+diff(y(x),x), -y(x)+diff(y(x),x) trying differential order: 1; missing the dependent variable checking if the LODE is of Euler type <- LODE of Euler type successful Euler equation successful trying differential order: 1; missing the dependent variable checking if the LODE is of Euler type <- LODE of Euler type successful Euler equation successful trying differential order: 1; missing the dependent variable checking if the LODE has constant coefficients <- constant coefficients successful <- solving the LCLM ode successful `
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 17
dsolve((x^2-2*x+2)*diff(y(x),x$3)-x^2*diff(y(x),x$2)+2*x*diff(y(x),x)-2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x +c_{2} x^{2}+c_{3} {\mathrm e}^{x} \]
✓ Solution by Mathematica
Time used: 0.124 (sec). Leaf size: 27
DSolve[(x^2-2*x+2)*y'''[x]-x^2*y''[x]+2*x*y'[x]-2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} \left (c_2 x^2+2 c_1 x+c_3 e^x\right ) \]