Internal problem ID [6792]
Book: Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam
Publishing Co. NY. 6th edition. 1981.
Section: CHAPTER 16. Nonlinear equations. Section 97. The p-discriminant equation.
EXERCISES Page 314
Problem number: 14.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational]
\[ \boxed {4 y^{3} {y^{\prime }}^{2}+4 x y^{\prime }+y=0} \]
✓ Solution by Maple
Time used: 0.313 (sec). Leaf size: 307
dsolve(4*y(x)^3*diff(y(x),x)^2+4*x*diff(y(x),x)+y(x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ \frac {\left (\int _{\textit {\_b}}^{x}\frac {-2 \textit {\_a} +\sqrt {-y \left (x \right )^{4}+\textit {\_a}^{2}}}{y \left (x \right )^{4}+3 \textit {\_a}^{2}}d \textit {\_a} \right )}{2}-\left (\int _{}^{y \left (x \right )}\frac {\left (1+\left (\textit {\_f}^{4}-\sqrt {-\textit {\_f}^{4}+x^{2}}\, x +x^{2}\right ) \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_f}^{4}+4 \sqrt {-\textit {\_f}^{4}+\textit {\_a}^{2}}\, \textit {\_a} -5 \textit {\_a}^{2}}{\sqrt {-\textit {\_f}^{4}+\textit {\_a}^{2}}\, \left (\textit {\_f}^{4}+3 \textit {\_a}^{2}\right )^{2}}d \textit {\_a} \right )\right ) \textit {\_f}^{3}}{\textit {\_f}^{4}-\sqrt {-\textit {\_f}^{4}+x^{2}}\, x +x^{2}}d \textit {\_f} \right )+c_{1} &= 0 \\ -\frac {\left (\int _{\textit {\_b}}^{x}\frac {2 \textit {\_a} +\sqrt {-y \left (x \right )^{4}+\textit {\_a}^{2}}}{y \left (x \right )^{4}+3 \textit {\_a}^{2}}d \textit {\_a} \right )}{2}-\left (\int _{}^{y \left (x \right )}\frac {\left (1+\left (\textit {\_f}^{4}+\sqrt {-\textit {\_f}^{4}+x^{2}}\, x +x^{2}\right ) \left (\int _{\textit {\_b}}^{x}\frac {-\textit {\_f}^{4}+5 \textit {\_a}^{2}+4 \sqrt {-\textit {\_f}^{4}+\textit {\_a}^{2}}\, \textit {\_a}}{\sqrt {-\textit {\_f}^{4}+\textit {\_a}^{2}}\, \left (\textit {\_f}^{4}+3 \textit {\_a}^{2}\right )^{2}}d \textit {\_a} \right )\right ) \textit {\_f}^{3}}{\textit {\_f}^{4}+\sqrt {-\textit {\_f}^{4}+x^{2}}\, x +x^{2}}d \textit {\_f} \right )+c_{1} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.284 (sec). Leaf size: 2815
DSolve[4*y[x]^3*(y'[x])^2+4*x*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
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