83.48.3 problem Ex 3 page 97

Internal problem ID [19595]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter VII. Exact differential equations.
Problem number : Ex 3 page 97
Date solved : Tuesday, January 28, 2025 at 02:01:14 PM
CAS classification : [[_3rd_order, _fully, _exact, _linear]]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 33

dsolve(diff(y(x),x$3)+cos(x)*diff(y(x),x$2)-2*sin(x)*diff(y(x),x)-y(x)*cos(x)=sin(2*x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.023 (sec). Leaf size: 75

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

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