Internal problem ID [15069]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact]
\[ \boxed {\frac {\sin \left (2 x \right )}{y}+\left (y-\frac {\sin \left (x \right )^{2}}{y^{2}}\right ) y^{\prime }=-x} \]
✓ Solution by Maple
Time used: 0.141 (sec). Leaf size: 401
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 \left (x \right ) &= \frac {\left (-108+108 \cos \left (2 x \right )+12 \sqrt {12 x^{6}+72 c_{1} x^{4}+144 x^{2} c_{1}^{2}+96 c_{1}^{3}+81-162 \cos \left (2 x \right )+81 \cos \left (2 x \right )^{2}}\right )^{\frac {2}{3}}-12 x^{2}-24 c_{1}}{6 \left (-108+108 \cos \left (2 x \right )+12 \sqrt {12 x^{6}+72 c_{1} x^{4}+144 x^{2} c_{1}^{2}+96 c_{1}^{3}+81-162 \cos \left (2 x \right )+81 \cos \left (2 x \right )^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (\frac {i \sqrt {3}}{12}+\frac {1}{12}\right ) \left (-108+108 \cos \left (2 x \right )+12 \sqrt {12 x^{6}+72 c_{1} x^{4}+144 x^{2} c_{1}^{2}+96 c_{1}^{3}+81-162 \cos \left (2 x \right )+81 \cos \left (2 x \right )^{2}}\right )^{\frac {2}{3}}+\left (i \sqrt {3}-1\right ) \left (x^{2}+2 c_{1} \right )}{\left (-108+108 \cos \left (2 x \right )+12 \sqrt {12 x^{6}+72 c_{1} x^{4}+144 x^{2} c_{1}^{2}+96 c_{1}^{3}+81-162 \cos \left (2 x \right )+81 \cos \left (2 x \right )^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (-108+108 \cos \left (2 x \right )+12 \sqrt {12 x^{6}+72 c_{1} x^{4}+144 x^{2} c_{1}^{2}+96 c_{1}^{3}+81-162 \cos \left (2 x \right )+81 \cos \left (2 x \right )^{2}}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )}{12}+\left (x^{2}+2 c_{1} \right ) \left (1+i \sqrt {3}\right )}{\left (-108+108 \cos \left (2 x \right )+12 \sqrt {12 x^{6}+72 c_{1} x^{4}+144 x^{2} c_{1}^{2}+96 c_{1}^{3}+81-162 \cos \left (2 x \right )+81 \cos \left (2 x \right )^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 6.373 (sec). Leaf size: 394
DSolve[( Sin[2*x]/y[x]+x )+( y[x]-Sin[x]^2/y[x]^2 )*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{2} \left (2 \sqrt {3} \sqrt {27 \sin ^4(x)+\left (x^2-c_1\right ){}^3}+9 \cos (2 x)-9\right ){}^{2/3}-2 \sqrt [3]{3} \left (x^2-c_1\right )}{6^{2/3} \sqrt [3]{2 \sqrt {3} \sqrt {27 \sin ^4(x)+\left (x^2-c_1\right ){}^3}+9 \cos (2 x)-9}} \\ y(x)\to \frac {6 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) \left (x^2-c_1\right )+i 6^{2/3} \left (\sqrt {3}+i\right ) \left (2 \sqrt {3} \sqrt {27 \sin ^4(x)+\left (x^2-c_1\right ){}^3}+9 \cos (2 x)-9\right ){}^{2/3}}{12 \sqrt [3]{3} \sqrt [3]{2 \sqrt {3} \sqrt {27 \sin ^4(x)+\left (x^2-c_1\right ){}^3}+9 \cos (2 x)-9}} \\ y(x)\to \frac {6 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) \left (x^2-c_1\right )-6^{2/3} \left (1+i \sqrt {3}\right ) \left (2 \sqrt {3} \sqrt {27 \sin ^4(x)+\left (x^2-c_1\right ){}^3}+9 \cos (2 x)-9\right ){}^{2/3}}{12 \sqrt [3]{3} \sqrt [3]{2 \sqrt {3} \sqrt {27 \sin ^4(x)+\left (x^2-c_1\right ){}^3}+9 \cos (2 x)-9}} \\ y(x)\to 0 \\ \end{align*}