28.1.111 problem 134

Internal problem ID [4417]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 134
Date solved : Monday, January 27, 2025 at 09:16:10 AM
CAS classification : [[_Abel, `2nd type`, `class B`]]

\begin{align*} 2 y \left (x \,{\mathrm e}^{x^{2}}+y \sin \left (x \right ) \cos \left (x \right )\right )+\left (2 \,{\mathrm e}^{x^{2}}+3 y \sin \left (x \right )^{2}\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.041 (sec). Leaf size: 657

dsolve((2*y(x)*(x*exp(x^2)+y(x)*sin(x)*cos(x)) )+(2*exp(x^2)+3*y(x)*sin(x)^2 )*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {2 \,{\mathrm e}^{x^{2}} \left (8 \,{\mathrm e}^{3 x^{2}}+3 \left (\sqrt {3}\, \sqrt {27}\, \sqrt {\left (\frac {16 \,{\mathrm e}^{3 x^{2}}}{27}+\left (-1+\cos \left (2 x \right )\right )^{2} c_{1} \right ) c_{1}}+9 \cos \left (2 x \right ) c_{1} -9 c_{1} \right ) \left (-1+\cos \left (2 x \right )\right )\right )^{{1}/{3}}+\left (8 \,{\mathrm e}^{3 x^{2}}+3 \left (\sqrt {3}\, \sqrt {27}\, \sqrt {\left (\frac {16 \,{\mathrm e}^{3 x^{2}}}{27}+\left (-1+\cos \left (2 x \right )\right )^{2} c_{1} \right ) c_{1}}+9 \cos \left (2 x \right ) c_{1} -9 c_{1} \right ) \left (-1+\cos \left (2 x \right )\right )\right )^{{2}/{3}}+4 \,{\mathrm e}^{2 x^{2}}}{\left (8 \,{\mathrm e}^{3 x^{2}}+3 \left (\sqrt {3}\, \sqrt {27}\, \sqrt {\left (\frac {16 \,{\mathrm e}^{3 x^{2}}}{27}+\left (-1+\cos \left (2 x \right )\right )^{2} c_{1} \right ) c_{1}}+9 \cos \left (2 x \right ) c_{1} -9 c_{1} \right ) \left (-1+\cos \left (2 x \right )\right )\right )^{{1}/{3}} \left (3 \cos \left (2 x \right )-3\right )} \\ y \left (x \right ) &= -\frac {-4 \,{\mathrm e}^{x^{2}} \left (8 \,{\mathrm e}^{3 x^{2}}+3 \left (\sqrt {3}\, \sqrt {27}\, \sqrt {\left (\frac {16 \,{\mathrm e}^{3 x^{2}}}{27}+\left (-1+\cos \left (2 x \right )\right )^{2} c_{1} \right ) c_{1}}+9 \cos \left (2 x \right ) c_{1} -9 c_{1} \right ) \left (-1+\cos \left (2 x \right )\right )\right )^{{1}/{3}}+\left (-4 i \sqrt {3}+4\right ) {\mathrm e}^{2 x^{2}}+\left (8 \,{\mathrm e}^{3 x^{2}}+3 \left (\sqrt {3}\, \sqrt {27}\, \sqrt {\left (\frac {16 \,{\mathrm e}^{3 x^{2}}}{27}+\left (-1+\cos \left (2 x \right )\right )^{2} c_{1} \right ) c_{1}}+9 \cos \left (2 x \right ) c_{1} -9 c_{1} \right ) \left (-1+\cos \left (2 x \right )\right )\right )^{{2}/{3}} \left (1+i \sqrt {3}\right )}{\left (8 \,{\mathrm e}^{3 x^{2}}+3 \left (\sqrt {3}\, \sqrt {27}\, \sqrt {\left (\frac {16 \,{\mathrm e}^{3 x^{2}}}{27}+\left (-1+\cos \left (2 x \right )\right )^{2} c_{1} \right ) c_{1}}+9 \cos \left (2 x \right ) c_{1} -9 c_{1} \right ) \left (-1+\cos \left (2 x \right )\right )\right )^{{1}/{3}} \left (-6+6 \cos \left (2 x \right )\right )} \\ y \left (x \right ) &= \frac {4 \,{\mathrm e}^{x^{2}} \left (8 \,{\mathrm e}^{3 x^{2}}+3 \left (\sqrt {3}\, \sqrt {27}\, \sqrt {\left (\frac {16 \,{\mathrm e}^{3 x^{2}}}{27}+\left (-1+\cos \left (2 x \right )\right )^{2} c_{1} \right ) c_{1}}+9 \cos \left (2 x \right ) c_{1} -9 c_{1} \right ) \left (-1+\cos \left (2 x \right )\right )\right )^{{1}/{3}}+\left (-4 i \sqrt {3}-4\right ) {\mathrm e}^{2 x^{2}}+\left (i \sqrt {3}-1\right ) \left (8 \,{\mathrm e}^{3 x^{2}}+3 \left (\sqrt {3}\, \sqrt {27}\, \sqrt {\left (\frac {16 \,{\mathrm e}^{3 x^{2}}}{27}+\left (-1+\cos \left (2 x \right )\right )^{2} c_{1} \right ) c_{1}}+9 \cos \left (2 x \right ) c_{1} -9 c_{1} \right ) \left (-1+\cos \left (2 x \right )\right )\right )^{{2}/{3}}}{\left (8 \,{\mathrm e}^{3 x^{2}}+3 \left (\sqrt {3}\, \sqrt {27}\, \sqrt {\left (\frac {16 \,{\mathrm e}^{3 x^{2}}}{27}+\left (-1+\cos \left (2 x \right )\right )^{2} c_{1} \right ) c_{1}}+9 \cos \left (2 x \right ) c_{1} -9 c_{1} \right ) \left (-1+\cos \left (2 x \right )\right )\right )^{{1}/{3}} \left (-6+6 \cos \left (2 x \right )\right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 64.239 (sec). Leaf size: 628

DSolve[(2*y[x]*(x*Exp[x^2]+y[x]*Sin[x]*Cos[x]) )+(2*Exp[x^2]+3*y[x]*Sin[x]^2 )*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {1}{12} \csc ^2(x) \left (4 e^{x^2}+\sqrt [3]{64 e^{3 x^2}+12 \sqrt {6} \sqrt {c_1 \sin ^4(x) \left (64 e^{3 x^2}-108 c_1 \cos (2 x)+27 c_1 \cos (4 x)+81 c_1\right )}-216 c_1 \cos (2 x)+54 c_1 \cos (4 x)+162 c_1}+\frac {16 e^{2 x^2}}{\sqrt [3]{64 e^{3 x^2}+12 \sqrt {6} \sqrt {c_1 \sin ^4(x) \left (64 e^{3 x^2}-108 c_1 \cos (2 x)+27 c_1 \cos (4 x)+81 c_1\right )}-216 c_1 \cos (2 x)+54 c_1 \cos (4 x)+162 c_1}}\right ) \\ y(x)\to -\frac {1}{24} \csc ^2(x) \left (8 e^{x^2}+i \left (\sqrt {3}+i\right ) \sqrt [3]{64 e^{3 x^2}+12 \sqrt {6} \sqrt {c_1 \sin ^4(x) \left (64 e^{3 x^2}-108 c_1 \cos (2 x)+27 c_1 \cos (4 x)+81 c_1\right )}-216 c_1 \cos (2 x)+54 c_1 \cos (4 x)+162 c_1}-\frac {16 i \left (\sqrt {3}-i\right ) e^{2 x^2}}{\sqrt [3]{64 e^{3 x^2}+12 \sqrt {6} \sqrt {c_1 \sin ^4(x) \left (64 e^{3 x^2}-108 c_1 \cos (2 x)+27 c_1 \cos (4 x)+81 c_1\right )}-216 c_1 \cos (2 x)+54 c_1 \cos (4 x)+162 c_1}}\right ) \\ y(x)\to -\frac {1}{24} \csc ^2(x) \left (8 e^{x^2}-i \left (\sqrt {3}-i\right ) \sqrt [3]{64 e^{3 x^2}+12 \sqrt {6} \sqrt {c_1 \sin ^4(x) \left (64 e^{3 x^2}-108 c_1 \cos (2 x)+27 c_1 \cos (4 x)+81 c_1\right )}-216 c_1 \cos (2 x)+54 c_1 \cos (4 x)+162 c_1}+\frac {16 i \left (\sqrt {3}+i\right ) e^{2 x^2}}{\sqrt [3]{64 e^{3 x^2}+12 \sqrt {6} \sqrt {c_1 \sin ^4(x) \left (64 e^{3 x^2}-108 c_1 \cos (2 x)+27 c_1 \cos (4 x)+81 c_1\right )}-216 c_1 \cos (2 x)+54 c_1 \cos (4 x)+162 c_1}}\right ) \\ \end{align*}