23.1.522 problem 512

Internal problem ID [5129]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 512
Date solved : Tuesday, September 30, 2025 at 11:43:37 AM
CAS classification : [[_homogeneous, `class D`], _Bernoulli]

\begin{align*} x y y^{\prime }&=a \,x^{3} \cos \left (x \right )+y^{2} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 30

ode:=x*y(x)*diff(y(x),x) = a*x^3*cos(x)+y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \sqrt {2 a \sin \left (x \right )+c_1}\, x \\ y &= -\sqrt {2 a \sin \left (x \right )+c_1}\, x \\ \end{align*}

✓ Mathematica. Time used: 0.178 (sec). Leaf size: 56

ode=x*y[x]*D[y[x],x]==a*x^3*Cos[x]+y[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -x \sqrt {2 \int _1^xa \cos (K[1])dK[1]+c_1}\\ y(x)&\to x \sqrt {2 \int _1^xa \cos (K[1])dK[1]+c_1} \end{align*}

✓ Sympy. Time used: 0.412 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*x**3*cos(x) + x*y(x)*Derivative(y(x), x) - y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - x \sqrt {C_{1} + 2 a \sin {\left (x \right )}}, \ y{\left (x \right )} = x \sqrt {C_{1} + 2 a \sin {\left (x \right )}}\right ] \]