40.14.11 problem 32

Internal problem ID [6782]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 19. Linear equations with variable coefficients (Misc. types). Supplemetary problems. Page 132
Problem number : 32
Date solved : Monday, January 27, 2025 at 02:30:25 PM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 49

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

\begin{align*} y &= -\frac {x}{2}-\frac {\sqrt {-4 c_1 x +5 x^{2}+4 c_2}}{2} \\ y &= -\frac {x}{2}+\frac {\sqrt {-4 c_1 x +5 x^{2}+4 c_2}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.645 (sec). Leaf size: 77

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

\begin{align*} y(x)\to -\frac {1}{2} x \left (1+\sqrt {\frac {1}{x^2}} \sqrt {5 x^2+4 c_2 x+4 c_1}\right ) \\ y(x)\to \frac {1}{2} x \left (-1+\sqrt {\frac {1}{x^2}} \sqrt {5 x^2+4 c_2 x+4 c_1}\right ) \\ \end{align*}