75.12.32 problem 306

Internal problem ID [16898]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 12. Miscellaneous problems. Exercises page 93
Problem number : 306
Date solved : Tuesday, January 28, 2025 at 09:40:10 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.911 (sec). Leaf size: 94

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

\[ y = {\mathrm e}^{\operatorname {RootOf}\left (-4 \sin \left (\textit {\_Z} \right ) \sin \left (\ln \left (x \right )\right ) {\mathrm e}^{\textit {\_Z}} x +4 \sin \left (\textit {\_Z} \right ) \cos \left (\ln \left (x \right )\right ) {\mathrm e}^{\textit {\_Z}} x -4 i x^{1+i} c_{1} +4 i x^{1-i} c_{1} -8 \sin \left (\textit {\_Z} \right ) {\mathrm e}^{\textit {\_Z}} c_{1} -4 \,{\mathrm e}^{2 \textit {\_Z}} \cos \left (\textit {\_Z} \right )^{2}-i x^{2-2 i}+i x^{2+2 i}-4 x^{1+i} c_{1} -4 x^{1-i} c_{1} +8 c_{1}^{2}+2 x^{2}+2 \,{\mathrm e}^{2 \textit {\_Z}}\right )} \]

✓ Solution by Mathematica

Time used: 0.328 (sec). Leaf size: 33

DSolve[Sin[Log[x]]-Cos[Log[y[x]]]*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\cos (\log (K[1]))dK[1]\&\right ]\left [\int _1^x\sin (\log (K[2]))dK[2]+c_1\right ] \]