60.4.22 problem 1470

Internal problem ID [11473]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1470
Date solved : Monday, January 27, 2025 at 11:22:29 PM
CAS classification : [[_3rd_order, _fully, _exact, _linear]]

\begin{align*} y^{\prime \prime \prime }-y^{\prime \prime } \sin \left (x \right )-2 y^{\prime } \cos \left (x \right )+y \sin \left (x \right )-\ln \left (x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 41

dsolve(diff(diff(diff(y(x),x),x),x)-diff(diff(y(x),x),x)*sin(x)-2*diff(y(x),x)*cos(x)+y(x)*sin(x)-ln(x)=0,y(x), singsol=all)
 

\[ y = \frac {\left (4 c_3 +\int \left (8 c_{1} x +4 c_{2} -3 x^{2}+2 \ln \left (x \right ) x^{2}\right ) {\mathrm e}^{\cos \left (x \right )}d x \right ) {\mathrm e}^{-\cos \left (x \right )}}{4} \]

✓ Solution by Mathematica

Time used: 0.068 (sec). Leaf size: 73

DSolve[-Log[x] + Sin[x]*y[x] - 2*Cos[x]*D[y[x],x] - Sin[x]*D[y[x],{x,2}] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\sin (K[1])dK[1]\right ) \left (\int _1^x\frac {1}{4} \exp \left (-\int _1^{K[2]}\sin (K[1])dK[1]\right ) \left (2 \log (K[2]) K[2]^2-3 K[2]^2+4 c_1 K[2]+4 c_2\right )dK[2]+c_3\right ) \]