83.22.8 problem 8

Internal problem ID [19262]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (E) at page 63
Problem number : 8
Date solved : Tuesday, January 28, 2025 at 01:17:22 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 86

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

\begin{align*} y \left (x \right ) &= \frac {1}{\sqrt {-x}} \\ y \left (x \right ) &= -\frac {1}{\sqrt {-x}} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\coth \left (-\frac {\ln \left (x \right )}{2}+\frac {c_{1}}{2}\right ) \sqrt {\operatorname {sech}\left (-\frac {\ln \left (x \right )}{2}+\frac {c_{1}}{2}\right )^{2} x}}{x} \\ y \left (x \right ) &= -\frac {\coth \left (-\frac {\ln \left (x \right )}{2}+\frac {c_{1}}{2}\right ) \sqrt {\operatorname {sech}\left (-\frac {\ln \left (x \right )}{2}+\frac {c_{1}}{2}\right )^{2} x}}{x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.493 (sec). Leaf size: 80

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

\begin{align*} y(x)\to -\frac {2 e^{\frac {c_1}{2}}}{-x+e^{c_1}} \\ y(x)\to \frac {2 e^{\frac {c_1}{2}}}{-x+e^{c_1}} \\ y(x)\to 0 \\ y(x)\to -\frac {i}{\sqrt {x}} \\ y(x)\to \frac {i}{\sqrt {x}} \\ \end{align*}