Internal problem ID [7209]
Internal file name [OUTPUT/6195_Sunday_June_05_2022_04_27_37_PM_96743745/index.tex]
Book: Own collection of miscellaneous problems
Section: section 3.0
Problem number: 19.
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.
\[ \boxed {x^{4} y^{\prime \prime \prime }+x^{3} y^{\prime \prime }+y^{\prime } x^{2}+y x=x} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{4} \left (\frac {d}{d x}y^{\prime \prime }\right )+\left (\frac {d}{d x}y^{\prime }\right ) x^{3}+y^{\prime } x^{2}+y x =x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d}{d x}y^{\prime \prime } \end {array} \]
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 3; linear nonhomogeneous with symmetry [0,1] trying high order linear exact nonhomogeneous trying differential order: 3; missing the dependent variable checking if the LODE is of Euler type <- LODE of Euler type successful Euler equation successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 223
dsolve(x^4*diff(y(x),x$3)+x^3*diff(y(x),x$2)+x^2*diff(y(x),x)+x*y(x)= x,y(x), singsol=all)
\[ y \left (x \right ) = c_{2} x^{\frac {\left (47-3 \sqrt {249}\right ) \left (188+12 \sqrt {249}\right )^{\frac {2}{3}}}{192}+\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}+\frac {2}{3}} \cos \left (\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}} \sqrt {3}\, \left (3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {83}-47 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}+16\right ) \ln \left (x \right )}{192}\right )+c_{3} x^{\frac {\left (47-3 \sqrt {249}\right ) \left (188+12 \sqrt {249}\right )^{\frac {2}{3}}}{192}+\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}+\frac {2}{3}} \sin \left (\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}} \sqrt {3}\, \left (3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {83}-47 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}+16\right ) \ln \left (x \right )}{192}\right )+x^{\frac {\left (188+12 \sqrt {249}\right )^{\frac {2}{3}} \left (-47+3 \sqrt {249}\right )}{96}-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}+\frac {2}{3}} c_{1} +1 \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 82
DSolve[x^4*y'''[x]+x^3*y''[x]+x^2*y'[x]+x*y[x]== x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,1\right ]}+c_3 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,3\right ]}+c_2 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,2\right ]}+1 \]