Internal problem ID [5442]
Internal file name [OUTPUT/4933_Tuesday_February_06_2024_10_14_25_PM_14447332/index.tex]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 19. Linear equations with variable coefficients (Misc. types). Supplemetary
problems. Page 132
Problem number: 35.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
Unable to parse ODE.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying 3rd order ODE linearizable_by_differentiation differential order: 3; trying a linearization to 4th order trying differential order: 3; missing variables trying differential order: 3; exact nonlinear -> Calling odsolve with the ODE`, (1/2)*_b(_a)^2+(diff(_b(_a), _a))^2-2*(diff(_b(_a), _a))*_b(_a)+(diff(diff(_b(_a), _a), _a))*_b(_a Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying a quadrature checking if the LODE has constant coefficients <- constant coefficients successful <- 2nd order, 2 integrating factors of the form mu(x,y) successful <- differential order: 3; exact nonlinear successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 51
dsolve(y(x)*diff(y(x),x$3)+3*diff(y(x),x)*diff(y(x),x$2)-2*y(x)*diff(y(x),x$2)-2*diff(y(x),x)^2+y(x)*diff(y(x),x)=exp(2*x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \sqrt {-2 c_{3} x \,{\mathrm e}^{x}+{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} c_{2} -2 c_{1}} \\ y \left (x \right ) &= -\sqrt {{\mathrm e}^{2 x}+\left (-2 c_{3} x +2 c_{2} \right ) {\mathrm e}^{x}-2 c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.387 (sec). Leaf size: 65
DSolve[y[x]*y'''[x]+3*y'[x]*y''[x]-2*y[x]*y''[x]-2*y'[x]^2+y[x]*y'[x]==Exp[2*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {e^{2 x}+e^x (c_3 x+2 c_2)+2 c_1} \\ y(x)\to \sqrt {e^{2 x}+e^x (c_3 x+2 c_2)+2 c_1} \\ \end{align*}