Internal problem ID [10010]
Internal file name [OUTPUT/8957_Monday_June_06_2022_06_03_16_AM_94243474/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1689 (book 6.98).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Unable to solve or complete the solution.
\[ \boxed {x^{4} y^{\prime \prime }-x^{2} \left (x +y^{\prime }\right ) y^{\prime }+4 y^{2}=0} \]
Maple trace
`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 differential order: 2; missing variables -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y)^2 --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y) trying 2nd order, integrating factors of the form mu(x,y)/(y)^n, only the singular cases integrating factor(s) found: 1/y(x)*exp(-x/y(x)^2)/diff(y(x),x)^2 attempting the computation of a related first integral... -> Calling odsolve with the ODE`, exp(-_b(_a)/_a^2)*(diff(_b(_a), _a))/_a+(2*exp(-_b(_a)/_a^2)*_a^2+c__1*_a^2+2*_b(_a)*exp(-_b(_a Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying homogeneous G <- homogeneous successful <- differential order: 2; exact nonlinear successful <- 2nd_order mu_xyp successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 32
dsolve(x^4*diff(diff(y(x),x),x)-x^2*(x+diff(y(x),x))*diff(y(x),x)+4*y(x)^2=0,y(x), singsol=all)
\[ y \left (x \right ) = \operatorname {RootOf}\left (-\ln \left (x \right )+c_{2} -\left (\int _{}^{\textit {\_Z}}\frac {1}{{\mathrm e}^{\textit {\_f}} c_{1} +4 \textit {\_f} +2}d \textit {\_f} \right )\right ) x^{2} \]
✓ Solution by Mathematica
Time used: 1.205 (sec). Leaf size: 189
DSolve[4*y[x]^2 - x^2*y'[x]*(x + y'[x]) + x^4*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\frac {1}{-e^{\frac {K[1]}{x^2}} c_1 x^2+2 x^2+4 K[1]}dK[1]-\int _1^x\left (\frac {K[2] \left (e^{\frac {y(x)}{K[2]^2}} c_1+2 \left (-\frac {y(x)}{K[2]^2}-1\right )\right )}{-e^{\frac {y(x)}{K[2]^2}} c_1 K[2]^2+2 K[2]^2+4 y(x)}+\int _1^{y(x)}-\frac {\frac {2 e^{\frac {K[1]}{K[2]^2}} c_1 K[1]}{K[2]}-2 e^{\frac {K[1]}{K[2]^2}} c_1 K[2]+4 K[2]}{\left (-e^{\frac {K[1]}{K[2]^2}} c_1 K[2]^2+2 K[2]^2+4 K[1]\right ){}^2}dK[1]\right )dK[2]=c_2,y(x)\right ] \]