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
:
Thursday, December 11, 2025 at 01:41:13 PMCAS
classification
:
[_separable]
2.1.13.1 Solved by factoring the differential equation Time used: 0.082 (sec)
\begin{align*}
3 t^{2} x-x t +\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 solver Solving 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 solver The ode
\begin{equation}
x^{\prime } = -\frac {3 t -1}{x t^{2} \left (x^{2}+3\right )}
\end{equation}
is separable as it can be written as
\begin{align*} x^{\prime }&= -\frac {3 t -1}{x t^{2} \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}{x \left (x^{2}+3\right )} \end{align*}
Integrating gives
\begin{align*}
\int { \frac {1}{g(x)} \,dx} &= \int { f(t) \,dt} \\
\int { x \left (x^{2}+3\right )\,dx} &= \int { -\frac {3 t -1}{t^{2}} \,dt} \\
\end{align*}
\[
\frac {\left (x^{2}+3\right )^{2}}{4}=-\frac {1}{t}+\ln \left (\frac {1}{t^{3}}\right )+c_1
\]
Figure 2.6: Slope field \(x^{\prime } x^{3} t^{2}+3 x^{\prime } x t^{2}+3 t -1 = 0\) 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: 6.959 (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 ) \[
\left [ x{\left (t \right )} = - \sqrt {-3 - \frac {\sqrt {t \left (C_{1} t - 12 t \log {\left (t \right )} + 9 t - 4\right )}}{t}}, \ x{\left (t \right )} = \sqrt {-3 - \frac {\sqrt {t \left (C_{1} t - 12 t \log {\left (t \right )} + 9 t - 4\right )}}{t}}, \ x{\left (t \right )} = - \sqrt {-3 + \frac {\sqrt {t \left (C_{1} t - 12 t \log {\left (t \right )} + 9 t - 4\right )}}{t}}, \ x{\left (t \right )} = \sqrt {-3 + \frac {\sqrt {t \left (C_{1} t - 12 t \log {\left (t \right )} + 9 t - 4\right )}}{t}}, \ x{\left (t \right )} = 0\right ]
\]