Internal problem ID [5441]
Internal file name [OUTPUT/4932_Tuesday_February_06_2024_10_14_25_PM_15270978/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: 34.
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`, (diff(diff(_b(_a), _a), _a))*_b(_a)^2*_a+2*_a*_b(_a)*(diff(_b(_a), _a))^2-2*(diff(_b(_a), _a))*_b( 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: 0.031 (sec). Leaf size: 104
dsolve(3*x*( y(x)^2* diff(y(x),x$3)+6*y(x)*diff(y(x),x)*diff(y(x),x$2)+2*diff(y(x),x)^3 )-3*y(x)* (y(x)*diff(y(x),x$2)+2* diff(y(x),x)^2 )=-2/x,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (8 \ln \left (x \right ) x -8 c_{3} x^{3}+\left (12 c_{1} -4\right ) x +8 c_{2} \right )^{\frac {1}{3}}}{2} \\ y \left (x \right ) &= -\frac {\left (8 \ln \left (x \right ) x -8 c_{3} x^{3}+\left (12 c_{1} -4\right ) x +8 c_{2} \right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{4} \\ y \left (x \right ) &= \frac {\left (8 \ln \left (x \right ) x -8 c_{3} x^{3}+\left (12 c_{1} -4\right ) x +8 c_{2} \right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{4} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.297 (sec). Leaf size: 121
DSolve[3*x*( y[x]^2* y'''[x]+6*y[x]*y'[x]*y''[x]+2*y'[x]^3 )-3*y[x]* (y[x]*y''[x]+2* y'[x]^2 )==-2/x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt [3]{-\frac {1}{6}} \sqrt [3]{6 c_3 x^3+6 x \log (x)+(3+9 c_2) x+2 c_1} \\ y(x)\to \sqrt [3]{c_3 x^3+x \log (x)+\frac {1}{2} (1+3 c_2) x+\frac {c_1}{3}} \\ y(x)\to (-1)^{2/3} \sqrt [3]{c_3 x^3+x \log (x)+\frac {1}{2} (1+3 c_2) x+\frac {c_1}{3}} \\ \end{align*}