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 : Sunday, March 30, 2025 at 03:16:30 AM
CAS classification : [[_homogeneous, `class G`], _dAlembert]

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

✓ Maple. Time used: 0.196 (sec). Leaf size: 67

ode:=y(x)+x*diff(y(x),x) = 4*diff(y(x),x)^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \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 &= \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*}

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

ode=y[x]+x*D[y[x],x]==4*Sqrt[D[y[x],x]]; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 1.507 (sec). Leaf size: 58

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x) + y(x) - 4*sqrt(Derivative(y(x), x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ \frac {\sqrt {- x y{\left (x \right )} + 4}}{2} - \log {\left (x \right )} + \log {\left (\sqrt {- x y{\left (x \right )} + 4} - 2 \right )} = C_{1}, \ - \frac {\sqrt {- x y{\left (x \right )} + 4}}{2} - \log {\left (x \right )} + \log {\left (\sqrt {- x y{\left (x \right )} + 4} + 2 \right )} = C_{1}\right ] \]