2.3.19 problem 19

Internal problem ID [8303]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 19
Date solved : Sunday, November 10, 2024 at 09:09:43 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

Solve

\begin{align*} x^{4} y^{\prime \prime \prime }+x^{3} y^{\prime \prime }+x^{2} y^{\prime }+x y&=x \end{align*}

Does not support this form of ODE for higher order. Terminating.

Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{4} \left (\frac {d^{3}}{d x^{3}}y \left (x \right )\right )+\left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right ) x^{3}+x^{2} \left (\frac {d}{d x}y \left (x \right )\right )+x y \left (x \right )=x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d^{3}}{d x^{3}}y \left (x \right ) \end {array} \]

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`
 

Maple dsolve solution

Solving time : 0.013 (sec)
Leaf size : 223

dsolve(x^4*diff(diff(diff(y(x),x),x),x)+x^3*diff(diff(y(x),x),x)+diff(y(x),x)*x^2+x*y(x) = x, 
       y(x),singsol=all)
 

\[ y = c_{2} x^{\frac {\left (47-3 \sqrt {249}\right ) \left (188+12 \sqrt {249}\right )^{{2}/{3}}}{192}+\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}+\frac {2}{3}} \cos \left (\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}} \sqrt {3}\, \left (3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}} \sqrt {3}\, \sqrt {83}-47 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+16\right ) \ln \left (x \right )}{192}\right )+c_3 \,x^{\frac {\left (47-3 \sqrt {249}\right ) \left (188+12 \sqrt {249}\right )^{{2}/{3}}}{192}+\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}+\frac {2}{3}} \sin \left (\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}} \sqrt {3}\, \left (3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}} \sqrt {3}\, \sqrt {83}-47 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+16\right ) \ln \left (x \right )}{192}\right )+x^{\frac {\left (-47+3 \sqrt {249}\right ) \left (188+12 \sqrt {249}\right )^{{2}/{3}}}{96}-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {2}{3}} c_{1} +1 \]

Mathematica DSolve solution

Solving time : 0.004 (sec)
Leaf size : 82

DSolve[{x^4*D[y[x],{x,3}]+x^3*D[y[x],{x,2}]+x^2*D[y[x],x]+x*y[x]== x,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,1\right ]}+c_3 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,3\right ]}+c_2 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,2\right ]}+1 \]