12.6.7 problem 7
Internal
problem
ID
[1686]
Book
:
Elementary
differential
equations
with
boundary
value
problems.
William
F.
Trench.
Brooks/Cole
2001
Section
:
Chapter
2,
First
order
equations.
Exact
equations.
Section
2.5
Page
79
Problem
number
:
7
Date
solved
:
Monday, January 27, 2025 at 05:27:19 AM
CAS
classification
:
[_exact]
\begin{align*} -2 y^{2} \sin \left (x \right )+3 y^{3}-2 x +\left (4 y \cos \left (x \right )+9 x y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.014 (sec). Leaf size: 636
dsolve((-2*y(x)^2*sin(x)+3*y(x)^3-2*x)+(4*y(x)*cos(x)+9*x*y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {\frac {\left (972 x^{4}-64 \cos \left (x \right )^{3}+36 \sqrt {3}\, \sqrt {\left (-x^{2}+c_1 \right ) \left (-243 x^{4}+32 \cos \left (x \right )^{3}+243 c_1 \,x^{2}\right )}\, x -972 c_1 \,x^{2}\right )^{{1}/{3}}}{2}+\frac {8 \cos \left (x \right )^{2}}{\left (972 x^{4}-64 \cos \left (x \right )^{3}+36 \sqrt {3}\, \sqrt {\left (-x^{2}+c_1 \right ) \left (-243 x^{4}+32 \cos \left (x \right )^{3}+243 c_1 \,x^{2}\right )}\, x -972 c_1 \,x^{2}\right )^{{1}/{3}}}-2 \cos \left (x \right )}{9 x} \\
y &= -\frac {\left (\left (972 x^{4}-64 \cos \left (x \right )^{3}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_1 \right )+243 \left (x^{2}-c_1 \right )^{2} x^{2}}\, x -972 c_1 \,x^{2}\right )^{{1}/{3}}+4 \cos \left (x \right )\right ) \left (i \left (\left (972 x^{4}-64 \cos \left (x \right )^{3}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_1 \right )+243 \left (x^{2}-c_1 \right )^{2} x^{2}}\, x -972 c_1 \,x^{2}\right )^{{1}/{3}}-4 \cos \left (x \right )\right ) \sqrt {3}+\left (972 x^{4}-64 \cos \left (x \right )^{3}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_1 \right )+243 \left (x^{2}-c_1 \right )^{2} x^{2}}\, x -972 c_1 \,x^{2}\right )^{{1}/{3}}+4 \cos \left (x \right )\right )}{36 x \left (972 x^{4}-64 \cos \left (x \right )^{3}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_1 \right )+243 \left (x^{2}-c_1 \right )^{2} x^{2}}\, x -972 c_1 \,x^{2}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \left (\left (972 x^{4}-64 \cos \left (x \right )^{3}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_1 \right )+243 \left (x^{2}-c_1 \right )^{2} x^{2}}\, x -972 c_1 \,x^{2}\right )^{{1}/{3}}-4 \cos \left (x \right )\right ) \sqrt {3}-\left (972 x^{4}-64 \cos \left (x \right )^{3}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_1 \right )+243 \left (x^{2}-c_1 \right )^{2} x^{2}}\, x -972 c_1 \,x^{2}\right )^{{1}/{3}}-4 \cos \left (x \right )\right ) \left (\left (972 x^{4}-64 \cos \left (x \right )^{3}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_1 \right )+243 \left (x^{2}-c_1 \right )^{2} x^{2}}\, x -972 c_1 \,x^{2}\right )^{{1}/{3}}+4 \cos \left (x \right )\right )}{36 x \left (972 x^{4}-64 \cos \left (x \right )^{3}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_1 \right )+243 \left (x^{2}-c_1 \right )^{2} x^{2}}\, x -972 c_1 \,x^{2}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 37.586 (sec). Leaf size: 469
DSolve[(-2*y[x]^2*Sin[x]+3*y[x]^3-2*x)+(4*y[x]*Cos[x]+9*x*y[x]^2)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {2^{2/3} \sqrt [3]{243 x^4+243 c_1 x^2+\sqrt {-256 \cos ^6(x)+\left (-16 \cos ^3(x)+243 x^2 \left (x^2+c_1\right )\right ){}^2}-16 \cos ^3(x)}+\frac {8 \sqrt [3]{2} \cos ^2(x)}{\sqrt [3]{243 x^4+243 c_1 x^2+\sqrt {-256 \cos ^6(x)+\left (-16 \cos ^3(x)+243 x^2 \left (x^2+c_1\right )\right ){}^2}-16 \cos ^3(x)}}-4 \cos (x)}{18 x} \\
y(x)\to \frac {i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{243 x^4+243 c_1 x^2+\sqrt {-256 \cos ^6(x)+\left (-16 \cos ^3(x)+243 x^2 \left (x^2+c_1\right )\right ){}^2}-16 \cos ^3(x)}-\frac {8 i \sqrt [3]{2} \left (\sqrt {3}-i\right ) \cos ^2(x)}{\sqrt [3]{243 x^4+243 c_1 x^2+\sqrt {-256 \cos ^6(x)+\left (-16 \cos ^3(x)+243 x^2 \left (x^2+c_1\right )\right ){}^2}-16 \cos ^3(x)}}-8 \cos (x)}{36 x} \\
y(x)\to -\frac {2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{243 x^4+243 c_1 x^2+\sqrt {-256 \cos ^6(x)+\left (-16 \cos ^3(x)+243 x^2 \left (x^2+c_1\right )\right ){}^2}-16 \cos ^3(x)}+\frac {8 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) \cos ^2(x)}{\sqrt [3]{243 x^4+243 c_1 x^2+\sqrt {-256 \cos ^6(x)+\left (-16 \cos ^3(x)+243 x^2 \left (x^2+c_1\right )\right ){}^2}-16 \cos ^3(x)}}+8 \cos (x)}{36 x} \\
\end{align*}