12.6.25 problem 25

Internal problem ID [1704]
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 : 25
Date solved : Monday, January 27, 2025 at 05:31:05 AM
CAS classification : [_exact, _rational]

\begin{align*} x^{3} y^{4}+x +\left (x^{4} y^{3}+y\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.011 (sec). Leaf size: 111

dsolve((x^3*y(x)^4+x)+(x^4*y(x)^3+y(x))*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y &= \frac {\sqrt {-1-\sqrt {-2 x^{6}-4 c_1 \,x^{4}+1}}}{x^{2}} \\ y &= \frac {\sqrt {-1+\sqrt {-2 x^{6}-4 c_1 \,x^{4}+1}}}{x^{2}} \\ y &= -\frac {\sqrt {-1-\sqrt {-2 x^{6}-4 c_1 \,x^{4}+1}}}{x^{2}} \\ y &= -\frac {\sqrt {-1+\sqrt {-2 x^{6}-4 c_1 \,x^{4}+1}}}{x^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 15.817 (sec). Leaf size: 135

DSolve[(x^3*y[x]^4+x)+(x^4*y[x]^3+y[x])*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sqrt {-\frac {1+\sqrt {-2 x^6+4 c_1 x^4+1}}{x^4}} \\ y(x)\to \sqrt {-\frac {1+\sqrt {-2 x^6+4 c_1 x^4+1}}{x^4}} \\ y(x)\to -\sqrt {\frac {-1+\sqrt {-2 x^6+4 c_1 x^4+1}}{x^4}} \\ y(x)\to \sqrt {\frac {-1+\sqrt {-2 x^6+4 c_1 x^4+1}}{x^4}} \\ \end{align*}