Internal problem ID [4464]
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]
\[ \boxed {y^{3}-\left (y^{2}+1-3 x y^{2}\right ) y^{\prime }=-4 x^{3}+\sin \left (x \right )} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 658
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 \left (x \right ) &= \frac {\left (2^{\frac {1}{3}} {\left (\left (-3 x^{4}-3 \cos \left (x \right )+\sqrt {\frac {\left (27 x -9\right ) \cos \left (x \right )^{2}+54 \left (x -\frac {1}{3}\right ) \left (x^{4}+c_{1} \right ) \cos \left (x \right )+27 x^{9}-9 x^{8}+54 c_{1} x^{5}-18 c_{1} x^{4}+27 c_{1}^{2} x -9 c_{1}^{2}-4}{3 x -1}}-3 c_{1} \right ) \left (3 x -1\right )^{2}\right )}^{\frac {2}{3}}+6 x -2\right ) 2^{\frac {1}{3}}}{{\left (\left (-3 x^{4}-3 \cos \left (x \right )+\sqrt {\frac {\left (27 x -9\right ) \cos \left (x \right )^{2}+54 \left (x -\frac {1}{3}\right ) \left (x^{4}+c_{1} \right ) \cos \left (x \right )+27 x^{9}-9 x^{8}+54 c_{1} x^{5}-18 c_{1} x^{4}+27 c_{1}^{2} x -9 c_{1}^{2}-4}{3 x -1}}-3 c_{1} \right ) \left (3 x -1\right )^{2}\right )}^{\frac {1}{3}} \left (6 x -2\right )} \\ y \left (x \right ) &= -\frac {\left (2^{\frac {1}{3}} \left (1+i \sqrt {3}\right ) {\left (-\left (3 x^{4}+3 \cos \left (x \right )-\sqrt {\frac {\left (27 x -9\right ) \cos \left (x \right )^{2}+54 \left (x -\frac {1}{3}\right ) \left (x^{4}+c_{1} \right ) \cos \left (x \right )+27 x^{9}-9 x^{8}+54 c_{1} x^{5}-18 c_{1} x^{4}+27 c_{1}^{2} x -9 c_{1}^{2}-4}{3 x -1}}+3 c_{1} \right ) \left (3 x -1\right )^{2}\right )}^{\frac {2}{3}}-6 \left (x -\frac {1}{3}\right ) \left (i \sqrt {3}-1\right )\right ) 2^{\frac {1}{3}}}{4 {\left (-\left (3 x^{4}+3 \cos \left (x \right )-\sqrt {\frac {\left (27 x -9\right ) \cos \left (x \right )^{2}+54 \left (x -\frac {1}{3}\right ) \left (x^{4}+c_{1} \right ) \cos \left (x \right )+27 x^{9}-9 x^{8}+54 c_{1} x^{5}-18 c_{1} x^{4}+27 c_{1}^{2} x -9 c_{1}^{2}-4}{3 x -1}}+3 c_{1} \right ) \left (3 x -1\right )^{2}\right )}^{\frac {1}{3}} \left (3 x -1\right )} \\ y \left (x \right ) &= \frac {\left (2^{\frac {1}{3}} \left (i \sqrt {3}-1\right ) {\left (-\left (3 x^{4}+3 \cos \left (x \right )-\sqrt {\frac {\left (27 x -9\right ) \cos \left (x \right )^{2}+54 \left (x -\frac {1}{3}\right ) \left (x^{4}+c_{1} \right ) \cos \left (x \right )+27 x^{9}-9 x^{8}+54 c_{1} x^{5}-18 c_{1} x^{4}+27 c_{1}^{2} x -9 c_{1}^{2}-4}{3 x -1}}+3 c_{1} \right ) \left (3 x -1\right )^{2}\right )}^{\frac {2}{3}}-6 \left (x -\frac {1}{3}\right ) \left (1+i \sqrt {3}\right )\right ) 2^{\frac {1}{3}}}{4 {\left (-\left (3 x^{4}+3 \cos \left (x \right )-\sqrt {\frac {\left (27 x -9\right ) \cos \left (x \right )^{2}+54 \left (x -\frac {1}{3}\right ) \left (x^{4}+c_{1} \right ) \cos \left (x \right )+27 x^{9}-9 x^{8}+54 c_{1} x^{5}-18 c_{1} x^{4}+27 c_{1}^{2} x -9 c_{1}^{2}-4}{3 x -1}}+3 c_{1} \right ) \left (3 x -1\right )^{2}\right )}^{\frac {1}{3}} \left (3 x -1\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.207 (sec). Leaf size: 682
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 (-27 x^6+18 x^5-3 x^4+\frac {1}{27} \sqrt {4 (9-27 x)^3+6561 (1-3 x)^4 \left (x^4+\cos (x)-c_1\right ){}^2}-27 x^2 \cos (x)+27 c_1 x^2+18 x \cos (x)-3 \cos (x)-18 c_1 x+3 c_1\right ){}^{2/3}+6 x-2}{2^{2/3} (3 x-1) \sqrt [3]{-27 x^6+18 x^5-3 x^4+\frac {1}{27} \sqrt {4 (9-27 x)^3+6561 (1-3 x)^4 \left (x^4+\cos (x)-c_1\right ){}^2}-27 x^2 \cos (x)+27 c_1 x^2+18 x \cos (x)-3 \cos (x)-18 c_1 x+3 c_1}} \\ y(x)\to \frac {9 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (-27 x^6+18 x^5-3 x^4+\frac {1}{27} \sqrt {4 (9-27 x)^3+6561 (1-3 x)^4 \left (x^4+\cos (x)-c_1\right ){}^2}-27 x^2 \cos (x)+27 c_1 x^2+18 x \cos (x)-3 \cos (x)-18 c_1 x+3 c_1\right ){}^{2/3}+2 \left (1+i \sqrt {3}\right ) (9-27 x)}{18\ 2^{2/3} (3 x-1) \sqrt [3]{-27 x^6+18 x^5-3 x^4+\frac {1}{27} \sqrt {4 (9-27 x)^3+6561 (1-3 x)^4 \left (x^4+\cos (x)-c_1\right ){}^2}-27 x^2 \cos (x)+27 c_1 x^2+18 x \cos (x)-3 \cos (x)-18 c_1 x+3 c_1}} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right )}{2^{2/3} \sqrt [3]{-27 x^6+18 x^5-3 x^4+\frac {1}{27} \sqrt {4 (9-27 x)^3+6561 (1-3 x)^4 \left (x^4+\cos (x)-c_1\right ){}^2}-27 x^2 \cos (x)+27 c_1 x^2+18 x \cos (x)-3 \cos (x)-18 c_1 x+3 c_1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-54 x^6+36 x^5-6 x^4+\frac {2}{27} \sqrt {4 (9-27 x)^3+6561 (1-3 x)^4 \left (x^4+\cos (x)-c_1\right ){}^2}-54 x^2 \cos (x)+54 c_1 x^2+36 x \cos (x)-6 \cos (x)-36 c_1 x+6 c_1}}{2\ 2^{2/3} (3 x-1)} \\ \end{align*}