60.1.388 problem 389

Internal problem ID [10402]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 389
Date solved : Monday, January 27, 2025 at 07:39:58 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}-\left (4 y+1\right ) y^{\prime }+\left (4 y+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.778 (sec). Leaf size: 137

dsolve(diff(y(x),x)^2-(4*y(x)+1)*diff(y(x),x)+(4*y(x)+1)*y(x) = 0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.204 (sec). Leaf size: 67

DSolve[y[x]*(1 + 4*y[x]) - (1 + 4*y[x])*D[y[x],x] + D[y[x],x]^2==0,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*}