2.1.24 Problem 25

Internal problem ID [10010]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 25
Date solved : Monday, January 26, 2026 at 08:47:39 PM
CAS classification : [_quadrature]

2.1.24.1 Solved by factoring the differential equation

Time used: 0.048 (sec)

\begin{align*} \left (y+x \right ) y^{\prime }&=0 \\ \end{align*}

Writing the ode as

\begin{align*} \left (y+x\right )\left (y^{\prime }\right )&=0 \end{align*}

Therefore we need to solve the following equations

\begin{align*} \tag{1} y+x &= 0 \\ \tag{2} y^{\prime } &= 0 \\ \end{align*}

Now each of the above equations is solved in turn.

Solving equation (1)

Entering zero order ode solverSolving for \(y\) from

\begin{align*} y+x = 0 \end{align*}

Solving gives

\begin{align*} y &= -x \\ \end{align*}

Solving equation (2)

Entering first order ode quadrature solverSince the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).

\begin{align*} \int {dy} &= \int {0\, dx} + c_1 \\ y &= c_1 \end{align*}

Figure 2.6: Isoclines for \(y^{\prime } = 0\)

Summary of solutions found

\begin{align*} y &= c_1 \\ y &= -x \\ \end{align*}

2.1.24.2 ✓ Maple. Time used: 0.001 (sec). Leaf size: 11

ode:=(x+y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -x \\ y &= c_1 \\ \end{align*}

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
<- 1st order linear successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (x +y \left (x \right )\right ) \left (\frac {d}{d x}y \left (x \right )\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=0 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (\frac {d}{d x}y \left (x \right )\right )d x =\int 0d x +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y \left (x \right )=\mathit {C1} \end {array} \]

2.1.24.3 ✓ Mathematica. Time used: 0.002 (sec). Leaf size: 14

ode=(x+y[x])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -x\\ y(x)&\to c_1 \end{align*}

2.1.24.4 ✓ Sympy. Time used: 0.175 (sec). Leaf size: 8

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

\[ \left [ y{\left (x \right )} = - x, \ y{\left (x \right )} = C_{1}\right ] \]

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0
 

classify_ode(ode,func=y(x)) 
 
('factorable', 'nth_algebraic', '1st_exact', '1st_linear', 'Bernoulli', '1st_power_series', 'lie_group', 'nth_algebraic_Integral', '1st_exact_Integral', '1st_linear_Integral', 'Bernoulli_Integral')