15.14.28 problem 30

Internal problem ID [3200]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 23, page 106
Problem number : 30
Date solved : Monday, January 27, 2025 at 07:25:54 AM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y^{\prime }&=x^{3}-\frac {\cos \left (2 x \right )}{2} \end{align*}

✓ Solution by Maple

Time used: 0.041 (sec). Leaf size: 1302

dsolve(diff(y(x),x$4)+2*diff(y(x),x$2)+diff(y(x),x)=x^3-1/2*cos(2*x),y(x), singsol=all)
 

\[ \text {Expression too large to display} \]

✓ Solution by Mathematica

Time used: 6.695 (sec). Leaf size: 1293

DSolve[D[y[x],{x,4}]+2*D[y[x],{x,2}]+D[y[x],x]==x^3-1/2*Cos[2*x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {e^{x \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,1\right ]} c_1}{\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,1\right ]}+c_4+\frac {i x \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]\right ) \left (4+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^2\right ) \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^3 x^3+4 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^2 x^2+12 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ] x+24\right )}{4 \sqrt {59} \left (-1+4 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^2\right )}+\frac {i x \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^3 x^3+3 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ] \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^2 x^3+3 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^2 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ] x^3+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^3 x^3-4 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^2 x^2-8 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ] \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ] x^2-4 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^2 x^2+12 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ] x+12 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ] x-24\right )}{4 \sqrt {59} \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right )^4}-\frac {i \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]\right ) \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^5 \sin (2 x)}{4 \sqrt {59} \left (-1+4 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^2\right )}-\frac {i \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right ) \sin (2 x)}{4 \sqrt {59} \left (-2 i+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right ) \left (2 i+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right )}-\frac {i \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^5 \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right ) \sin (2 x)}{4 \sqrt {59} \left (1-4 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^2\right )}-\frac {i \cos (2 x) \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]\right ) \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^4}{2 \sqrt {59} \left (-1+4 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^2\right )}+\frac {i \cos (2 x) \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right )}{2 \sqrt {59} \left (-2 i+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right ) \left (2 i+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right )}+\frac {e^{x \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]} c_3}{\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]}+\frac {i x \left (4+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^2\right ) \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^3 x^3+4 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^2 x^2+12 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ] x+24\right ) \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right )}{4 \sqrt {59} \left (1-4 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^2\right )}-\frac {i \cos (2 x) \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^4 \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right )}{2 \sqrt {59} \left (1-4 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^2\right )}+\frac {e^{x \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]} c_2}{\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]} \]