75.7.6 problem 180

Internal problem ID [16799]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 7, Total differential equations. The integrating factor. Exercises page 61
Problem number : 180
Date solved : Tuesday, January 28, 2025 at 09:28:54 AM
CAS classification : [_exact]

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

✓ Solution by Maple

Time used: 0.106 (sec). Leaf size: 397

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

\begin{align*} y &= \frac {\left (108 \cos \left (2 x \right )-108+12 \sqrt {12 x^{6}+72 c_{1} x^{4}+144 c_{1}^{2} x^{2}+96 c_{1}^{3}+81 \cos \left (2 x \right )^{2}-162 \cos \left (2 x \right )+81}\right )^{{2}/{3}}-12 x^{2}-24 c_{1}}{6 \left (108 \cos \left (2 x \right )-108+12 \sqrt {12 x^{6}+72 c_{1} x^{4}+144 c_{1}^{2} x^{2}+96 c_{1}^{3}+81 \cos \left (2 x \right )^{2}-162 \cos \left (2 x \right )+81}\right )^{{1}/{3}}} \\ y &= -\frac {\left (\frac {i \sqrt {3}}{12}+\frac {1}{12}\right ) \left (108 \cos \left (2 x \right )-108+12 \sqrt {12 x^{6}+72 c_{1} x^{4}+144 c_{1}^{2} x^{2}+96 c_{1}^{3}+81 \cos \left (2 x \right )^{2}-162 \cos \left (2 x \right )+81}\right )^{{2}/{3}}+\left (x^{2}+2 c_{1} \right ) \left (i \sqrt {3}-1\right )}{\left (108 \cos \left (2 x \right )-108+12 \sqrt {12 x^{6}+72 c_{1} x^{4}+144 c_{1}^{2} x^{2}+96 c_{1}^{3}+81 \cos \left (2 x \right )^{2}-162 \cos \left (2 x \right )+81}\right )^{{1}/{3}}} \\ y &= \frac {\frac {\left (108 \cos \left (2 x \right )-108+12 \sqrt {12 x^{6}+72 c_{1} x^{4}+144 c_{1}^{2} x^{2}+96 c_{1}^{3}+81 \cos \left (2 x \right )^{2}-162 \cos \left (2 x \right )+81}\right )^{{2}/{3}} \left (i \sqrt {3}-1\right )}{12}+\left (x^{2}+2 c_{1} \right ) \left (1+i \sqrt {3}\right )}{\left (108 \cos \left (2 x \right )-108+12 \sqrt {12 x^{6}+72 c_{1} x^{4}+144 c_{1}^{2} x^{2}+96 c_{1}^{3}+81 \cos \left (2 x \right )^{2}-162 \cos \left (2 x \right )+81}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.353 (sec). Leaf size: 74

DSolve[( Sin[2*x]/y[x]+x  )+( y[x]-Sin[x]^2/y[x]^2 )*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^x\left (2 K[1]+\frac {2 \sin (2 K[1])}{y(x)}\right )dK[1]+\int _1^{y(x)}\left (\frac {\cos (2 x)-1}{K[2]^2}+2 K[2]-\int _1^x-\frac {2 \sin (2 K[1])}{K[2]^2}dK[1]\right )dK[2]=c_1,y(x)\right ] \]