67.2.51 problem Problem 19(d)
Internal
problem
ID
[14008]
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)
Date
solved
:
Tuesday, January 28, 2025 at 06:12:28 AM
CAS
classification
:
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]
\begin{align*} y y^{\prime \prime } \sin \left (x \right )+\left (\sin \left (x \right ) y^{\prime }+\cos \left (x \right ) y\right ) y^{\prime }&=\cos \left (x \right ) \end{align*}
✓ Solution by Maple
Time used: 0.477 (sec). Leaf size: 793
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 &= \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 (1+{\mathrm e}^{i x}\right )^{2}\right )+1\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i x} \left (1+{\mathrm e}^{i x}\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 (1+{\mathrm e}^{i x}\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+3 \,{\mathrm e}^{4 i x}-3 \,{\mathrm e}^{2 i x}-{\mathrm e}^{6 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+3 \,{\mathrm e}^{4 i x}-3 \,{\mathrm e}^{2 i x}-{\mathrm e}^{6 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+3 \,{\mathrm e}^{2 i x}-3 \,{\mathrm e}^{4 i x}+{\mathrm e}^{6 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 (1+{\mathrm e}^{i x}\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 (1+{\mathrm e}^{i x}\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 )}}{-6-18 \,{\mathrm e}^{4 i x}+18 \,{\mathrm e}^{2 i x}+6 \,{\mathrm e}^{6 i x}} \\
y &= \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 (1+{\mathrm e}^{i x}\right )^{2}\right )+1\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i x} \left (1+{\mathrm e}^{i x}\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 (1+{\mathrm e}^{i x}\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+3 \,{\mathrm e}^{4 i x}-3 \,{\mathrm e}^{2 i x}-{\mathrm e}^{6 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+3 \,{\mathrm e}^{4 i x}-3 \,{\mathrm e}^{2 i x}-{\mathrm e}^{6 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+3 \,{\mathrm e}^{2 i x}-3 \,{\mathrm e}^{4 i x}+{\mathrm e}^{6 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 (1+{\mathrm e}^{i x}\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 (1+{\mathrm e}^{i x}\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+18 \,{\mathrm e}^{4 i x}-18 \,{\mathrm e}^{2 i x}-6 \,{\mathrm e}^{6 i x}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.145 (sec). Leaf size: 92
DSolve[y[x]*D[y[x],{x,2}]*Sin[x]+ ( D[y[x],x]*Sin[x]+y[x]*Cos[x] )*D[y[x],x]==Cos[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\sqrt {2} \sqrt {\int _1^x-\csc (K[2]) \left (c_1+\int _1^{K[2]}-\cos (K[1])dK[1]\right )dK[2]+c_2} \\
y(x)\to \sqrt {2} \sqrt {\int _1^x-\csc (K[2]) \left (c_1+\int _1^{K[2]}-\cos (K[1])dK[1]\right )dK[2]+c_2} \\
\end{align*}