8.5.40 problem 40

Internal problem ID [768]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.6, Substitution methods and exact equations. Page 74
Problem number : 40
Date solved : Monday, January 27, 2025 at 03:04:19 AM
CAS classification : [_exact]

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

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 153

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

\[ y = \arctan \left (-\frac {c_1 \operatorname {RootOf}\left (\textit {\_Z}^{4} {\mathrm e}^{2 x}+2 x \,{\mathrm e}^{x} \textit {\_Z}^{3}+\left (c_1^{2}+x^{2}-{\mathrm e}^{2 x}\right ) \textit {\_Z}^{2}-2 x \,{\mathrm e}^{x} \textit {\_Z} -x^{2}\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{4} {\mathrm e}^{2 x}+2 x \,{\mathrm e}^{x} \textit {\_Z}^{3}+\left (c_1^{2}+x^{2}-{\mathrm e}^{2 x}\right ) \textit {\_Z}^{2}-2 x \,{\mathrm e}^{x} \textit {\_Z} -x^{2}\right ) {\mathrm e}^{x}+x}, \operatorname {RootOf}\left (\textit {\_Z}^{4} {\mathrm e}^{2 x}+2 x \,{\mathrm e}^{x} \textit {\_Z}^{3}+\left (c_1^{2}+x^{2}-{\mathrm e}^{2 x}\right ) \textit {\_Z}^{2}-2 x \,{\mathrm e}^{x} \textit {\_Z} -x^{2}\right )\right ) \]

✓ Solution by Mathematica

Time used: 60.852 (sec). Leaf size: 5539

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

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