Internal problem ID [3956]
Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 2. Special types of differential equations of the first kind. Lesson 9
Problem number: Exact Differential equations. Exercise 9.13, page 79.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact]
Solve \begin {gather*} \boxed {4 x^{3}-\sin \relax (x )+y^{3}-\left (y^{2}+1-3 x y^{2}\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 1162
dsolve((4*x^3-sin(x)+y(x)^3)-(y(x)^2+1-3*x*y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (\left (-12 x^{4}-12 \cos \relax (x )+4 \sqrt {\frac {27 x^{9}-9 x^{8}+54 \cos \relax (x ) x^{5}+54 x^{5} c_{1}-18 x^{4} \cos \relax (x )-18 x^{4} c_{1}+27 x \left (\cos ^{2}\relax (x )\right )+54 c_{1} x \cos \relax (x )+27 x c_{1}^{2}-9 \left (\cos ^{2}\relax (x )\right )-18 c_{1} \cos \relax (x )-9 c_{1}^{2}-4}{3 x -1}}-12 c_{1}\right ) \left (3 x -1\right )^{2}\right )^{\frac {1}{3}}}{6 x -2}+\frac {2}{\left (\left (-12 x^{4}-12 \cos \relax (x )+4 \sqrt {\frac {27 x^{9}-9 x^{8}+54 \cos \relax (x ) x^{5}+54 x^{5} c_{1}-18 x^{4} \cos \relax (x )-18 x^{4} c_{1}+27 x \left (\cos ^{2}\relax (x )\right )+54 c_{1} x \cos \relax (x )+27 x c_{1}^{2}-9 \left (\cos ^{2}\relax (x )\right )-18 c_{1} \cos \relax (x )-9 c_{1}^{2}-4}{3 x -1}}-12 c_{1}\right ) \left (3 x -1\right )^{2}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (\left (-12 x^{4}-12 \cos \relax (x )+4 \sqrt {\frac {27 x^{9}-9 x^{8}+54 \cos \relax (x ) x^{5}+54 x^{5} c_{1}-18 x^{4} \cos \relax (x )-18 x^{4} c_{1}+27 x \left (\cos ^{2}\relax (x )\right )+54 c_{1} x \cos \relax (x )+27 x c_{1}^{2}-9 \left (\cos ^{2}\relax (x )\right )-18 c_{1} \cos \relax (x )-9 c_{1}^{2}-4}{3 x -1}}-12 c_{1}\right ) \left (3 x -1\right )^{2}\right )^{\frac {1}{3}}}{4 \left (3 x -1\right )}-\frac {1}{\left (\left (-12 x^{4}-12 \cos \relax (x )+4 \sqrt {\frac {27 x^{9}-9 x^{8}+54 \cos \relax (x ) x^{5}+54 x^{5} c_{1}-18 x^{4} \cos \relax (x )-18 x^{4} c_{1}+27 x \left (\cos ^{2}\relax (x )\right )+54 c_{1} x \cos \relax (x )+27 x c_{1}^{2}-9 \left (\cos ^{2}\relax (x )\right )-18 c_{1} \cos \relax (x )-9 c_{1}^{2}-4}{3 x -1}}-12 c_{1}\right ) \left (3 x -1\right )^{2}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (\left (-12 x^{4}-12 \cos \relax (x )+4 \sqrt {\frac {27 x^{9}-9 x^{8}+54 \cos \relax (x ) x^{5}+54 x^{5} c_{1}-18 x^{4} \cos \relax (x )-18 x^{4} c_{1}+27 x \left (\cos ^{2}\relax (x )\right )+54 c_{1} x \cos \relax (x )+27 x c_{1}^{2}-9 \left (\cos ^{2}\relax (x )\right )-18 c_{1} \cos \relax (x )-9 c_{1}^{2}-4}{3 x -1}}-12 c_{1}\right ) \left (3 x -1\right )^{2}\right )^{\frac {1}{3}}}{6 x -2}-\frac {2}{\left (\left (-12 x^{4}-12 \cos \relax (x )+4 \sqrt {\frac {27 x^{9}-9 x^{8}+54 \cos \relax (x ) x^{5}+54 x^{5} c_{1}-18 x^{4} \cos \relax (x )-18 x^{4} c_{1}+27 x \left (\cos ^{2}\relax (x )\right )+54 c_{1} x \cos \relax (x )+27 x c_{1}^{2}-9 \left (\cos ^{2}\relax (x )\right )-18 c_{1} \cos \relax (x )-9 c_{1}^{2}-4}{3 x -1}}-12 c_{1}\right ) \left (3 x -1\right )^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (\left (-12 x^{4}-12 \cos \relax (x )+4 \sqrt {\frac {27 x^{9}-9 x^{8}+54 \cos \relax (x ) x^{5}+54 x^{5} c_{1}-18 x^{4} \cos \relax (x )-18 x^{4} c_{1}+27 x \left (\cos ^{2}\relax (x )\right )+54 c_{1} x \cos \relax (x )+27 x c_{1}^{2}-9 \left (\cos ^{2}\relax (x )\right )-18 c_{1} \cos \relax (x )-9 c_{1}^{2}-4}{3 x -1}}-12 c_{1}\right ) \left (3 x -1\right )^{2}\right )^{\frac {1}{3}}}{4 \left (3 x -1\right )}-\frac {1}{\left (\left (-12 x^{4}-12 \cos \relax (x )+4 \sqrt {\frac {27 x^{9}-9 x^{8}+54 \cos \relax (x ) x^{5}+54 x^{5} c_{1}-18 x^{4} \cos \relax (x )-18 x^{4} c_{1}+27 x \left (\cos ^{2}\relax (x )\right )+54 c_{1} x \cos \relax (x )+27 x c_{1}^{2}-9 \left (\cos ^{2}\relax (x )\right )-18 c_{1} \cos \relax (x )-9 c_{1}^{2}-4}{3 x -1}}-12 c_{1}\right ) \left (3 x -1\right )^{2}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (\left (-12 x^{4}-12 \cos \relax (x )+4 \sqrt {\frac {27 x^{9}-9 x^{8}+54 \cos \relax (x ) x^{5}+54 x^{5} c_{1}-18 x^{4} \cos \relax (x )-18 x^{4} c_{1}+27 x \left (\cos ^{2}\relax (x )\right )+54 c_{1} x \cos \relax (x )+27 x c_{1}^{2}-9 \left (\cos ^{2}\relax (x )\right )-18 c_{1} \cos \relax (x )-9 c_{1}^{2}-4}{3 x -1}}-12 c_{1}\right ) \left (3 x -1\right )^{2}\right )^{\frac {1}{3}}}{6 x -2}-\frac {2}{\left (\left (-12 x^{4}-12 \cos \relax (x )+4 \sqrt {\frac {27 x^{9}-9 x^{8}+54 \cos \relax (x ) x^{5}+54 x^{5} c_{1}-18 x^{4} \cos \relax (x )-18 x^{4} c_{1}+27 x \left (\cos ^{2}\relax (x )\right )+54 c_{1} x \cos \relax (x )+27 x c_{1}^{2}-9 \left (\cos ^{2}\relax (x )\right )-18 c_{1} \cos \relax (x )-9 c_{1}^{2}-4}{3 x -1}}-12 c_{1}\right ) \left (3 x -1\right )^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 49.714 (sec). Leaf size: 567
DSolve[(4*x^3-Sin[x]+y[x]^3)-(y[x]^2+1-3*x*y[x]^2)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{2} \left (-3 (1-3 x)^2 \left (x^4-c_1\right )+\frac {1}{27} \sqrt {4 (9-27 x)^3+6561 (1-3 x)^4 \left (x^4+\cos (x)-c_1\right ){}^2}-3 (1-3 x)^2 \cos (x)\right ){}^{2/3}+6 x-2}{2^{2/3} (3 x-1) \sqrt [3]{-3 (1-3 x)^2 \left (x^4-c_1\right )+\frac {1}{27} \sqrt {4 (9-27 x)^3+6561 (1-3 x)^4 \left (x^4+\cos (x)-c_1\right ){}^2}-3 (1-3 x)^2 \cos (x)}} \\ y(x)\to \frac {9 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (-3 (1-3 x)^2 \left (x^4-c_1\right )+\frac {1}{27} \sqrt {4 (9-27 x)^3+6561 (1-3 x)^4 \left (x^4+\cos (x)-c_1\right ){}^2}-3 (1-3 x)^2 \cos (x)\right ){}^{2/3}+\left (2+2 i \sqrt {3}\right ) (9-27 x)}{18\ 2^{2/3} (3 x-1) \sqrt [3]{-3 (1-3 x)^2 \left (x^4-c_1\right )+\frac {1}{27} \sqrt {4 (9-27 x)^3+6561 (1-3 x)^4 \left (x^4+\cos (x)-c_1\right ){}^2}-3 (1-3 x)^2 \cos (x)}} \\ y(x)\to \frac {i \left (2 \left (\sqrt {3}+i\right )-\frac {\sqrt [3]{2} \left (\sqrt {3}-i\right ) \left (-3 (1-3 x)^2 \left (x^4-c_1\right )+\frac {1}{27} \sqrt {4 (9-27 x)^3+6561 (1-3 x)^4 \left (x^4+\cos (x)-c_1\right ){}^2}-3 (1-3 x)^2 \cos (x)\right ){}^{2/3}}{3 x-1}\right )}{2\ 2^{2/3} \sqrt [3]{-3 (1-3 x)^2 \left (x^4-c_1\right )+\frac {1}{27} \sqrt {4 (9-27 x)^3+6561 (1-3 x)^4 \left (x^4+\cos (x)-c_1\right ){}^2}-3 (1-3 x)^2 \cos (x)}} \\ \end{align*}