28.1.88 problem 91

Internal problem ID [4394]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 91
Date solved : Monday, January 27, 2025 at 09:14:00 AM
CAS classification : [[_homogeneous, `class G`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.074 (sec). Leaf size: 67

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

\begin{align*} y \left (x \right ) &= \frac {8 \sqrt {\frac {\operatorname {LambertW}\left (-\frac {c_{1} x}{2}\right )^{2}}{x^{2}}}\, x -4 \operatorname {LambertW}\left (-\frac {c_{1} x}{2}\right )^{2}}{x} \\ y \left (x \right ) &= \frac {8 \sqrt {\frac {\operatorname {LambertW}\left (\frac {c_{1} x}{2}\right )^{2}}{x^{2}}}\, x -4 \operatorname {LambertW}\left (\frac {c_{1} x}{2}\right )^{2}}{x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.125 (sec). Leaf size: 94

DSolve[y[x]+x*D[y[x],x]==4*Sqrt[D[y[x],x]],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {Solve}\left [\frac {2 e^{-\frac {1}{2} \sqrt {4-x y(x)}} \left (-2 \sqrt {4-x y(x)}-4\right )}{y(x)}&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {2 e^{\frac {1}{2} \sqrt {4-x y(x)}} \left (2 \sqrt {4-x y(x)}-4\right )}{y(x)}&=c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}