83.23.24 problem 24

Internal problem ID [19305]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter V. Singular solutions. Exercise V at page 76
Problem number : 24
Date solved : Tuesday, January 28, 2025 at 01:28:43 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}&=\left (4 y+1\right ) \left (y^{\prime }-y\right ) \end{align*}

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 137

dsolve(diff(y(x),x)^2=(4*y(x)+1)*(diff(y(x),x)-y(x)),y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -{\frac {1}{4}} \\ y \left (x \right ) &= -\frac {\sqrt {-{\mathrm e}^{-2 x} c_{1}}\, {\mathrm e}^{2 x}+c_{1}}{c_{1} \sqrt {-{\mathrm e}^{-2 x} c_{1}}} \\ y \left (x \right ) &= \frac {-\sqrt {-{\mathrm e}^{-2 x} c_{1}}\, {\mathrm e}^{2 x}+c_{1}}{\sqrt {-{\mathrm e}^{-2 x} c_{1}}\, c_{1}} \\ y \left (x \right ) &= \frac {-\sqrt {-{\mathrm e}^{-2 x} c_{1}}\, {\mathrm e}^{2 x}+c_{1}}{\sqrt {-{\mathrm e}^{-2 x} c_{1}}\, c_{1}} \\ y \left (x \right ) &= -\frac {\sqrt {-{\mathrm e}^{-2 x} c_{1}}\, {\mathrm e}^{2 x}+c_{1}}{c_{1} \sqrt {-{\mathrm e}^{-2 x} c_{1}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.193 (sec). Leaf size: 67

DSolve[D[y[x],x]^2==(4*y[x]+1)*(D[y[x],x]-y[x]),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{4} e^{x-4 c_1} \left (e^x+2 e^{2 c_1}\right ) \\ y(x)\to \frac {1}{4} e^{x+2 c_1} \left (-2+e^{x+2 c_1}\right ) \\ y(x)\to -\frac {1}{4} \\ y(x)\to 0 \\ \end{align*}