53.3.7 problem 9

Internal problem ID [8469]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 99. Clairaut equation. EXERCISES Page 320
Problem number : 9
Date solved : Monday, January 27, 2025 at 04:06:28 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} x^{4} {y^{\prime }}^{2}+2 x^{3} y y^{\prime }-4&=0 \end{align*}

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 47

dsolve(x^4*diff(y(x),x)^2+2*x^3*y(x)*diff(y(x),x)-4=0,y(x), singsol=all)
 

\begin{align*} y &= -\frac {2 i}{x} \\ y &= \frac {2 i}{x} \\ y &= \frac {2 \sinh \left (-\ln \left (x \right )+c_{1} \right )}{x} \\ y &= -\frac {2 \sinh \left (-\ln \left (x \right )+c_{1} \right )}{x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.805 (sec). Leaf size: 71

DSolve[x^4*(D[y[x],x])^2+2*x^3*y[x]*D[y[x],x]-4==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {4 e^{c_1}}{x^2}-\frac {e^{-c_1}}{4} \\ y(x)\to \frac {e^{-c_1}}{4}-\frac {4 e^{c_1}}{x^2} \\ y(x)\to -\frac {2 i}{x} \\ y(x)\to \frac {2 i}{x} \\ \end{align*}