9.7.6 problem 6

Internal problem ID [2987]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 11, page 45
Problem number : 6
Date solved : Tuesday, September 30, 2025 at 06:13:57 AM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }-x y&=\sqrt {y}\, x \,{\mathrm e}^{x^{2}} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 23

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

\[ \sqrt {y}-\frac {{\mathrm e}^{x^{2}}}{3}-{\mathrm e}^{\frac {x^{2}}{4}} c_1 = 0 \]

✓ Mathematica. Time used: 0.188 (sec). Leaf size: 34

ode=D[y[x],x]-x*y[x]==Sqrt[y[x]]*x*Exp[x^2]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{9} e^{\frac {x^2}{2}} \left (e^{\frac {3 x^2}{4}}+3 c_1\right ){}^2 \end{align*}

✓ Sympy. Time used: 0.276 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*sqrt(y(x))*exp(x**2) - x*y(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1}^{2} e^{\frac {x^{2}}{2}} + \frac {2 C_{1} e^{\frac {5 x^{2}}{4}}}{3} + \frac {e^{2 x^{2}}}{9} \]