Internal problem ID [9850]
Internal file name [OUTPUT/8795_Monday_June_06_2022_05_30_09_AM_82119685/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1526.
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.
\[ \boxed {x^{2} \left (x^{4}+2 x^{2}+2 x +1\right ) y^{\prime \prime \prime }-\left (2 x^{6}+3 x^{4}-6 x^{2}-6 x -1\right ) y^{\prime \prime }+\left (x^{6}-6 x^{3}-15 x^{2}-12 x -2\right ) y^{\prime }+\left (x^{4}+4 x^{3}+8 x^{2}+6 x +1\right ) y=0} \] 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 -y(x)+diff(y(x),x), (-1-1/x)*y(x)+diff(y(x),x), 1/x^2*y(x)+diff(y(x),x) trying differential order: 1; missing the dependent variable checking if the LODE has constant coefficients <- constant coefficients successful trying differential order: 1; missing the dependent variable checking if the LODE is of Euler type checking if the LODE is of Euler type exponential solutions successful <- differential factorization successful trying differential order: 1; missing the dependent variable checking if the LODE is of Euler type checking if the LODE is of Euler type exponential solutions successful <- differential factorization successful <- solving the LCLM ode successful `
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 19
dsolve(x^2*(x^4+2*x^2+2*x+1)*diff(diff(diff(y(x),x),x),x)-(2*x^6+3*x^4-6*x^2-6*x-1)*diff(diff(y(x),x),x)+(x^6-6*x^3-15*x^2-12*x-2)*diff(y(x),x)+(x^4+4*x^3+8*x^2+6*x+1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{2} {\mathrm e}^{\frac {1}{x}}+{\mathrm e}^{x} \left (c_{3} x +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 130.169 (sec). Leaf size: 25
DSolve[(1 + 6*x + 8*x^2 + 4*x^3 + x^4)*y[x] + (-2 - 12*x - 15*x^2 - 6*x^3 + x^6)*y'[x] - (-1 - 6*x - 6*x^2 + 3*x^4 + 2*x^6)*y''[x] + x^2*(1 + 2*x + 2*x^2 + x^4)*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^x (c_2 x+c_1)+c_3 e^{\frac {1}{x}} \]