Internal problem ID [12272]
Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A.
Dobrushkin. CRC Press 2015
Section: Chapter 4, Second and Higher Order Linear Differential Equations. Problems page
221
Problem number: Problem 19(d).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]
\[ \boxed {y y^{\prime \prime } \sin \left (x \right )+\left (y^{\prime } \sin \left (x \right )+y \cos \left (x \right )\right ) y^{\prime }=\cos \left (x \right )} \]
✓ Solution by Maple
Time used: 0.282 (sec). Leaf size: 898
dsolve(y(x)*diff(y(x),x$2)*sin(x)+ ( diff(y(x),x)*sin(x)+y(x)*cos(x) )*diff(y(x),x)=cos(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\sqrt {12}\, \sqrt {\left (-{\mathrm e}^{2 i x}+1\right )^{3} \left (-\frac {i}{3}+\frac {{\mathrm e}^{3 i x} \pi \left (\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{i x}-1\right )^{2}\right )-1\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i x} \left ({\mathrm e}^{i x}-1\right )^{2}\right )}{2}-\frac {{\mathrm e}^{3 i x} \pi \left (\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{i x}+1\right )^{2}\right )+1\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i x} \left ({\mathrm e}^{i x}+1\right )^{2}\right )}{2}+\frac {{\mathrm e}^{3 i x} \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{i x}-1\right )^{2}\right )}{2}-\frac {{\mathrm e}^{3 i x} \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{i x}+1\right )^{2}\right )}{2}+6 \left (1-{\mathrm e}^{6 i x}+3 \,{\mathrm e}^{4 i x}-3 \,{\mathrm e}^{2 i x}\right ) \left (\int \cos \left (x \right ) \cot \left (x \right )^{4} \operatorname {arctanh}\left (\cos \left (x \right )\right )d x \right )+\frac {21 \left (1-{\mathrm e}^{6 i x}+3 \,{\mathrm e}^{4 i x}-3 \,{\mathrm e}^{2 i x}\right ) \left (\int \cot \left (x \right ) \csc \left (x \right )^{3} \operatorname {arctanh}\left (\cos \left (x \right )\right )d x \right )}{4}+12 \left (-1+{\mathrm e}^{6 i x}-3 \,{\mathrm e}^{4 i x}+3 \,{\mathrm e}^{2 i x}\right ) \left (\int \cot \left (x \right )^{3} \csc \left (x \right ) \operatorname {arctanh}\left (\cos \left (x \right )\right )d x \right )-{\mathrm e}^{3 i x} \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{i x}-1\right )\right )+{\mathrm e}^{3 i x} \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{i x}+1\right )\right )-2 i {\mathrm e}^{3 i x} \ln \left ({\mathrm e}^{i x}+1\right )+2 i {\mathrm e}^{3 i x} \ln \left ({\mathrm e}^{i x}-1\right )+{\mathrm e}^{2 i x} \left (i+18 \left (-c_{2} +\sin \left (x \right )\right ) \operatorname {arctanh}\left (\cos \left (x \right )\right )+18 c_{1} \right )+18 \left (\left (c_{2} -\sin \left (x \right )\right ) \operatorname {arctanh}\left (\cos \left (x \right )\right )-c_{1} \right ) {\mathrm e}^{4 i x}+6 \left (\left (-c_{2} +\sin \left (x \right )\right ) \operatorname {arctanh}\left (\cos \left (x \right )\right )+c_{1} \right ) {\mathrm e}^{6 i x}+6 \left (c_{2} -\sin \left (x \right )\right ) \operatorname {arctanh}\left (\cos \left (x \right )\right )-6 c_{1} \right )}}{-6+6 \,{\mathrm e}^{6 i x}-18 \,{\mathrm e}^{4 i x}+18 \,{\mathrm e}^{2 i x}} \\ y \left (x \right ) &= \frac {\sqrt {12}\, \sqrt {\left (-{\mathrm e}^{2 i x}+1\right )^{3} \left (-\frac {i}{3}+\frac {{\mathrm e}^{3 i x} \pi \left (\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{i x}-1\right )^{2}\right )-1\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i x} \left ({\mathrm e}^{i x}-1\right )^{2}\right )}{2}-\frac {{\mathrm e}^{3 i x} \pi \left (\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{i x}+1\right )^{2}\right )+1\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i x} \left ({\mathrm e}^{i x}+1\right )^{2}\right )}{2}+\frac {{\mathrm e}^{3 i x} \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{i x}-1\right )^{2}\right )}{2}-\frac {{\mathrm e}^{3 i x} \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{i x}+1\right )^{2}\right )}{2}+6 \left (1-{\mathrm e}^{6 i x}+3 \,{\mathrm e}^{4 i x}-3 \,{\mathrm e}^{2 i x}\right ) \left (\int \cos \left (x \right ) \cot \left (x \right )^{4} \operatorname {arctanh}\left (\cos \left (x \right )\right )d x \right )+\frac {21 \left (1-{\mathrm e}^{6 i x}+3 \,{\mathrm e}^{4 i x}-3 \,{\mathrm e}^{2 i x}\right ) \left (\int \cot \left (x \right ) \csc \left (x \right )^{3} \operatorname {arctanh}\left (\cos \left (x \right )\right )d x \right )}{4}+12 \left (-1+{\mathrm e}^{6 i x}-3 \,{\mathrm e}^{4 i x}+3 \,{\mathrm e}^{2 i x}\right ) \left (\int \cot \left (x \right )^{3} \csc \left (x \right ) \operatorname {arctanh}\left (\cos \left (x \right )\right )d x \right )-{\mathrm e}^{3 i x} \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{i x}-1\right )\right )+{\mathrm e}^{3 i x} \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{i x}+1\right )\right )-2 i {\mathrm e}^{3 i x} \ln \left ({\mathrm e}^{i x}+1\right )+2 i {\mathrm e}^{3 i x} \ln \left ({\mathrm e}^{i x}-1\right )+{\mathrm e}^{2 i x} \left (i+18 \left (-c_{2} +\sin \left (x \right )\right ) \operatorname {arctanh}\left (\cos \left (x \right )\right )+18 c_{1} \right )+18 \left (\left (c_{2} -\sin \left (x \right )\right ) \operatorname {arctanh}\left (\cos \left (x \right )\right )-c_{1} \right ) {\mathrm e}^{4 i x}+6 \left (\left (-c_{2} +\sin \left (x \right )\right ) \operatorname {arctanh}\left (\cos \left (x \right )\right )+c_{1} \right ) {\mathrm e}^{6 i x}+6 \left (c_{2} -\sin \left (x \right )\right ) \operatorname {arctanh}\left (\cos \left (x \right )\right )-6 c_{1} \right )}}{-6+6 \,{\mathrm e}^{6 i x}-18 \,{\mathrm e}^{4 i x}+18 \,{\mathrm e}^{2 i x}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.159 (sec). Leaf size: 50
DSolve[y[x]*y''[x]*Sin[x]+ ( y'[x]*Sin[x]+y[x]*Cos[x] )*y'[x]==Cos[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {2} \sqrt {c_1 \text {arctanh}(\cos (x))+x+c_2} \\ y(x)\to \sqrt {2} \sqrt {c_1 \text {arctanh}(\cos (x))+x+c_2} \\ \end{align*}