Internal problem ID [14299]
Internal file name [OUTPUT/13980_Saturday_March_09_2024_04_42_33_PM_51637518/index.tex]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 2. First Order Equations. Exercises 2.4, page 57
Problem number: 12.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {3 y^{2} y^{\prime }=1} \]
Integrating both sides gives \begin {align*} \int 3 y^{2}d y &= t +c_{1}\\ y^{3}&=t +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\left (t +c_{1} \right )^{\frac {1}{3}}\\ y_2&=-\frac {\left (t +c_{1} \right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (t +c_{1} \right )^{\frac {1}{3}}}{2}\\ y_3&=-\frac {\left (t +c_{1} \right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (t +c_{1} \right )^{\frac {1}{3}}}{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \left (t +c_{1} \right )^{\frac {1}{3}} \\ \tag{2} y &= -\frac {\left (t +c_{1} \right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (t +c_{1} \right )^{\frac {1}{3}}}{2} \\ \tag{3} y &= -\frac {\left (t +c_{1} \right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (t +c_{1} \right )^{\frac {1}{3}}}{2} \\ \end{align*}
Verification of solutions
\[ y = \left (t +c_{1} \right )^{\frac {1}{3}} \] Verified OK.
\[ y = -\frac {\left (t +c_{1} \right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (t +c_{1} \right )^{\frac {1}{3}}}{2} \] Verified OK.
\[ y = -\frac {\left (t +c_{1} \right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (t +c_{1} \right )^{\frac {1}{3}}}{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 3 y^{2} y^{\prime }=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{2} y^{\prime }=\frac {1}{3} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int y^{2} y^{\prime }d t =\int \frac {1}{3}d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{3}}{3}=\frac {t}{3}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\left (3 c_{1} +t \right )^{\frac {1}{3}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 45
dsolve(-1+3*y(t)^2*diff(y(t),t)=0,y(t), singsol=all)
\begin{align*} y \left (t \right ) &= \left (t +c_{1} \right )^{\frac {1}{3}} \\ y \left (t \right ) &= -\frac {\left (t +c_{1} \right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{2} \\ y \left (t \right ) &= \frac {\left (t +c_{1} \right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.036 (sec). Leaf size: 56
DSolve[-1+3*y[t]^2*y'[t]==0,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \sqrt [3]{t+3 c_1} \\ y(t)\to -\sqrt [3]{-1} \sqrt [3]{t+3 c_1} \\ y(t)\to (-1)^{2/3} \sqrt [3]{t+3 c_1} \\ \end{align*}