Internal problem ID [2264]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 25, page 112
Problem number: 15.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _exact, _linear, _nonhomogeneous]]
\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+7 x^{3} y^{\prime \prime \prime }+9 x^{2} y^{\prime \prime }-6 x y^{\prime }-6 y=\cos \left (\ln \left (x \right )\right )} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 394
dsolve(x^4*diff(y(x),x$4)+7*x^3*diff(y(x),x$3)+9*x^2*diff(y(x),x$2)-6*x*diff(y(x),x)-6*y(x)=cos(ln(x)),y(x), singsol=all)
\[ y = \frac {\sin \left (\sqrt {2}\, \ln \left (x \right )\right ) c_{4}}{x}+\frac {\cos \left (\sqrt {2}\, \ln \left (x \right )\right ) c_{3}}{x}+\frac {33 \left (\left (-2-4 i+\left (-1+3 i\right ) \sqrt {2}\right ) \cos \left (\sqrt {2}\, \ln \left (x \right )\right )-\left (-4+2 i+\left (3+i\right ) \sqrt {2}\right ) \sin \left (\sqrt {2}\, \ln \left (x \right )\right )\right ) x^{-i \sqrt {2}+1-i}+33 \left (\left (-2+4 i+\left (1+3 i\right ) \sqrt {2}\right ) \cos \left (\sqrt {2}\, \ln \left (x \right )\right )+\left (-4-2 i+\left (-3+i\right ) \sqrt {2}\right ) \sin \left (\sqrt {2}\, \ln \left (x \right )\right )\right ) x^{-i \sqrt {2}+1+i}+33 \left (\left (-2-4 i+\left (1-3 i\right ) \sqrt {2}\right ) \cos \left (\sqrt {2}\, \ln \left (x \right )\right )-\left (4-2 i+\left (3+i\right ) \sqrt {2}\right ) \sin \left (\sqrt {2}\, \ln \left (x \right )\right )\right ) x^{i \sqrt {2}+1-i}+\left (120 c_{2} \left (2-3 i \sqrt {2}\right ) x^{3-i \sqrt {2}}+120 c_{2} \left (2+3 i \sqrt {2}\right ) x^{3+i \sqrt {2}}-440 x^{-i \sqrt {2}} c_{1} -440 c_{1} x^{i \sqrt {2}}-33 x^{i \sqrt {2}+1+i} \left (\left (1+3 i\right ) \sqrt {2}+2-4 i\right )\right ) \cos \left (\sqrt {2}\, \ln \left (x \right )\right )+33 \left (\frac {80 \left (i+\frac {3 \sqrt {2}}{2}\right ) c_{2} x^{3-i \sqrt {2}}}{11}-\frac {80 \left (i-\frac {3 \sqrt {2}}{2}\right ) c_{2} x^{3+i \sqrt {2}}}{11}-\frac {40 i c_{1} x^{-i \sqrt {2}}}{3}+\frac {40 i c_{1} x^{i \sqrt {2}}}{3}+x^{i \sqrt {2}+1+i} \left (4+2 i+\left (-3+i\right ) \sqrt {2}\right )\right ) \sin \left (\sqrt {2}\, \ln \left (x \right )\right )}{5280 x} \]
✓ Solution by Mathematica
Time used: 0.318 (sec). Leaf size: 62
DSolve[x^4*y''''[x]+7*x^3*y'''[x]+9*x^2*y''[x]-6*x*y'[x]-6*y[x]==Cos[Log[x]],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_4 x^2+\frac {c_3}{x}-\frac {1}{10} \sin (\log (x))-\frac {1}{20} \cos (\log (x))+\frac {c_2 \cos \left (\sqrt {2} \log (x)\right )}{x}+\frac {c_1 \sin \left (\sqrt {2} \log (x)\right )}{x} \]