2.1.13 Problem 3 (i)

Internal problem ID [19671]
Book : Elementary Differential Equations. By R.L.E. Schwarzenberger. Chapman and Hall. London. First Edition (1969)
Section : Chapter 3. Solutions of first-order equations. Exercises at page 47
Problem number : 3 (i)
Date solved : Wednesday, January 28, 2026 at 10:44:13 AM
CAS classification : [_separable]

2.1.13.1 Solved by factoring the differential equation

Time used: 0.388 (sec)

\begin{align*} 3 t^{2} x-t x+\left (3 t^{3} x^{2}+t^{3} x^{4}\right ) x^{\prime }&=0 \\ \end{align*}

Writing the ode as

\begin{align*} \left (x\right )\left (x^{\prime } x^{3} t^{2}+3 x^{\prime } x t^{2}+3 t -1\right )&=0 \end{align*}

Therefore we need to solve the following equations

\begin{align*} \tag{1} x &= 0 \\ \tag{2} x^{\prime } x^{3} t^{2}+3 x^{\prime } x t^{2}+3 t -1 &= 0 \\ \end{align*}

Now each of the above equations is solved in turn.

Solving equation (1)

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

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

Solving gives

\begin{align*} x &= 0 \\ \end{align*}

Solving equation (2)

Entering first order ode separable solverThe ode

\begin{equation} x^{\prime } = -\frac {3 t -1}{t^{2} x \left (x^{2}+3\right )} \end{equation}

is separable as it can be written as

\begin{align*} x^{\prime }&= -\frac {3 t -1}{t^{2} x \left (x^{2}+3\right )}\\ &= f(t) g(x) \end{align*}

Where

\begin{align*} f(t) &= -\frac {3 t -1}{t^{2}}\\ g(x) &= \frac {1}{\left (x^{2}+3\right ) x} \end{align*}

Integrating gives

\begin{align*} \int { \frac {1}{g(x)} \,dx} &= \int { f(t) \,dt} \\ \int { \left (x^{2}+3\right ) x\,dx} &= \int { -\frac {3 t -1}{t^{2}} \,dt} \\ \end{align*}

\[ \frac {\left (x^{2}+3\right )^{2}}{4}=-\frac {1}{t}+\ln \left (\frac {1}{t^{3}}\right )+c_1 \]

Direction field \(x^{\prime } x^{3} t^{2}+3 x^{\prime } x t^{2}+3 t -1 = 0\)	Isoclines for \(x^{\prime } x^{3} t^{2}+3 x^{\prime } x t^{2}+3 t -1 = 0\)

Summary of solutions found

\begin{align*} \frac {\left (x^{2}+3\right )^{2}}{4} &= -\frac {1}{t}+\ln \left (\frac {1}{t^{3}}\right )+c_1 \\ x &= 0 \\ \end{align*}

2.1.13.2 ✓ Maple. Time used: 0.007 (sec). Leaf size: 139

ode:=3*t^2*x(t)-t*x(t)+(3*t^3*x(t)^2+t^3*x(t)^4)*diff(x(t),t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\begin{align*} x &= 0 \\ x &= \frac {\sqrt {-3 t^{2}+2 t \sqrt {-t \left (1+3 \ln \left (t \right ) t +c_1 t \right )}}}{t} \\ x &= \frac {\sqrt {-3 t^{2}-2 t \sqrt {-t \left (1+3 \ln \left (t \right ) t +c_1 t \right )}}}{t} \\ x &= -\frac {\sqrt {-3 t^{2}+2 t \sqrt {-t \left (1+3 \ln \left (t \right ) t +c_1 t \right )}}}{t} \\ x &= -\frac {\sqrt {-3 t^{2}-2 t \sqrt {-t \left (1+3 \ln \left (t \right ) t +c_1 t \right )}}}{t} \\ \end{align*}

Maple trace

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

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 3 t^{2} x \left (t \right )-t x \left (t \right )+\left (3 t^{3} x \left (t \right )^{2}+t^{3} x \left (t \right )^{4}\right ) \left (\frac {d}{d t}x \left (t \right )\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d t}x \left (t \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d t}x \left (t \right )=\frac {-3 t^{2} x \left (t \right )+t x \left (t \right )}{3 t^{3} x \left (t \right )^{2}+t^{3} x \left (t \right )^{4}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d t}x \left (t \right )\right ) x \left (t \right ) \left (x \left (t \right )^{2}+3\right )=-\frac {3 t -1}{t^{2}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \left (\frac {d}{d t}x \left (t \right )\right ) x \left (t \right ) \left (x \left (t \right )^{2}+3\right )d t =\int -\frac {3 t -1}{t^{2}}d t +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\left (x \left (t \right )^{2}+3\right )^{2}}{4}=-\frac {1}{t}-3 \ln \left (t \right )+\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \left (t \right ) \\ {} & {} & \left \{x \left (t \right )=\frac {\sqrt {t \left (-3 t +2 \sqrt {-3 \ln \left (t \right ) t^{2}+\mathit {C1} \,t^{2}-t}\right )}}{t}, x \left (t \right )=\frac {\sqrt {-t \left (3 t +2 \sqrt {-3 \ln \left (t \right ) t^{2}+\mathit {C1} \,t^{2}-t}\right )}}{t}, x \left (t \right )=-\frac {\sqrt {t \left (-3 t +2 \sqrt {-3 \ln \left (t \right ) t^{2}+\mathit {C1} \,t^{2}-t}\right )}}{t}, x \left (t \right )=-\frac {\sqrt {-t \left (3 t +2 \sqrt {-3 \ln \left (t \right ) t^{2}+\mathit {C1} \,t^{2}-t}\right )}}{t}\right \} \end {array} \]

2.1.13.3 ✓ Mathematica. Time used: 9.261 (sec). Leaf size: 157

ode=(3*t^2*x[t]-t*x[t])+(3*t^3*x[t]^2+t^3*x[t]^4)*D[x[t],t]==0; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 0\\ x(t)&\to -\sqrt {-3-\frac {\sqrt {9 t-12 t \log (t)+4 c_1 t-4}}{\sqrt {t}}}\\ x(t)&\to \sqrt {-3-\frac {\sqrt {9 t-12 t \log (t)+4 c_1 t-4}}{\sqrt {t}}}\\ x(t)&\to -\sqrt {-3+\frac {\sqrt {9 t-12 t \log (t)+4 c_1 t-4}}{\sqrt {t}}}\\ x(t)&\to \sqrt {-3+\frac {\sqrt {9 t-12 t \log (t)+4 c_1 t-4}}{\sqrt {t}}}\\ x(t)&\to 0 \end{align*}

2.1.13.4 ✓ Sympy. Time used: 6.872 (sec). Leaf size: 126

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(3*t**2*x(t) - t*x(t) + (t**3*x(t)**4 + 3*t**3*x(t)**2)*Derivative(x(t), t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ \left [ x{\left (t \right )} = - \sqrt {-3 - \frac {\sqrt {t \left (C_{1} t - 12 t \log {\left (t \right )} + 9 t - 4\right )}}{t}}, \ x{\left (t \right )} = \sqrt {-3 - \frac {\sqrt {t \left (C_{1} t - 12 t \log {\left (t \right )} + 9 t - 4\right )}}{t}}, \ x{\left (t \right )} = - \sqrt {-3 + \frac {\sqrt {t \left (C_{1} t - 12 t \log {\left (t \right )} + 9 t - 4\right )}}{t}}, \ x{\left (t \right )} = \sqrt {-3 + \frac {\sqrt {t \left (C_{1} t - 12 t \log {\left (t \right )} + 9 t - 4\right )}}{t}}, \ x{\left (t \right )} = 0\right ] \]

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=x(t)) 
 
('factorable', 'separable', 'lie_group', 'separable_Integral')