Internal problem ID [9895]
Internal file name [OUTPUT/8842_Monday_June_06_2022_05_36_49_AM_36004170/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1573.
ODE order: 4.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_high_order, _fully, _exact, _linear]]
Unable to solve or complete the solution.
\[ \boxed {\left ({\mathrm e}^{x}+2 x \right ) y^{\prime \prime \prime \prime }+4 \left ({\mathrm e}^{x}+2\right ) y^{\prime \prime \prime }+6 \,{\mathrm e}^{x} y^{\prime \prime }+4 \,{\mathrm e}^{x} y^{\prime }+y \,{\mathrm e}^{x}=\frac {1}{x^{5}}} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left ({\mathrm e}^{x}+2 x \right ) \left (\frac {d}{d x}\frac {d}{d x}y^{\prime \prime }\right )+4 \left ({\mathrm e}^{x}+2\right ) \left (\frac {d}{d x}y^{\prime \prime }\right )+6 \,{\mathrm e}^{x} \left (\frac {d}{d x}y^{\prime }\right )+4 \,{\mathrm e}^{x} y^{\prime }+y \,{\mathrm e}^{x}=\frac {1}{x^{5}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\prime \prime } \end {array} \]
Maple trace
`Methods for high order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable <- high order exact linear fully integrable successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 41
dsolve((exp(x)+2*x)*diff(diff(diff(diff(y(x),x),x),x),x)+4*(exp(x)+2)*diff(diff(diff(y(x),x),x),x)+6*exp(x)*diff(diff(y(x),x),x)+4*exp(x)*diff(y(x),x)+y(x)*exp(x)-1/x^5=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {24 c_{1} x^{4}+24 c_{2} x^{3}+24 x^{2} c_{3} +24 c_{4} x +1}{24 \left ({\mathrm e}^{x}+2 x \right ) x} \]
✓ Solution by Mathematica
Time used: 0.152 (sec). Leaf size: 48
DSolve[-x^(-5) + E^x*y[x] + 4*E^x*y'[x] + 6*E^x*y''[x] + 4*(2 + E^x)*Derivative[3][y][x] + (E^x + 2*x)*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {24 c_4 x^4+24 c_3 x^3+24 c_2 x^2+24 c_1 x+1}{48 x^2+24 e^x x} \]