Internal problem ID [11326]
Internal file name [OUTPUT/10312_Tuesday_December_27_2022_04_06_16_AM_28478648/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 60. Exact equation. Integrating factor. Page 139
Problem number: Ex 5.
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`, _b(_a)*(diff(diff(_b(_a), _a), _a))*_a^3+_a^3*(diff(_b(_a), _a))^2+6*_b(_a)*(diff(_b(_a), _a))*_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 checking if the LODE is of Euler type <- LODE of Euler type successful <- 2nd order, 2 integrating factors of the form mu(x,y) successful <- differential order: 3; exact nonlinear successful`
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 56
dsolve(2*x^3*y(x)*diff(y(x),x$3)+6*x^3*diff(y(x),x)*diff(y(x),x$2)+18*x^2*y(x)*diff(y(x),x$2)+18*x^2*diff(y(x),x)^2+36*x*y(x)*diff(y(x),x)+6*y(x)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\sqrt {-x \left (c_{1} x^{2}+2 c_{2} x -2 c_{3} \right )}}{x^{2}} \\ y \left (x \right ) &= -\frac {\sqrt {-x \left (c_{1} x^{2}+2 c_{2} x -2 c_{3} \right )}}{x^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.389 (sec). Leaf size: 60
DSolve[2*x^3*y[x]*y'''[x]+6*x^3*y'[x]*y''[x]+18*x^2*y[x]*y''[x]+18*x^2*y'[x]^2+36*x*y[x]*y'[x]+6*y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {c_1 x^2+c_3 x+2 c_2}}{x^{3/2}} \\ y(x)\to \frac {\sqrt {c_1 x^2+c_3 x+2 c_2}}{x^{3/2}} \\ \end{align*}