Internal problem ID [14222]
Internal file name [OUTPUT/13903_Saturday_March_09_2024_03_57_51_PM_24203072/index.tex]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 2. First Order Equations. Exercises 2.2, page 39
Problem number: 61 (b).
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]
\[ \boxed {y^{\prime }-\left (3 y+1\right )^{4}=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{\left (3 y +1\right )^{4}}d y &= x +c_{1}\\ -\frac {1}{9 \left (3 y +1\right )^{3}}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {\left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{9 x +9 c_{1}}-\frac {1}{3}\\ y_2&=-\frac {\left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{18 \left (x +c_{1} \right )}-\frac {i \sqrt {3}\, \left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{18 \left (x +c_{1} \right )}-\frac {1}{3}\\ y_3&=-\frac {\left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{18 \left (x +c_{1} \right )}+\frac {i \sqrt {3}\, \left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{18 x +18 c_{1}}-\frac {1}{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{9 x +9 c_{1}}-\frac {1}{3} \\ \tag{2} y &= -\frac {\left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{18 \left (x +c_{1} \right )}-\frac {i \sqrt {3}\, \left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{18 \left (x +c_{1} \right )}-\frac {1}{3} \\ \tag{3} y &= -\frac {\left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{18 \left (x +c_{1} \right )}+\frac {i \sqrt {3}\, \left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{18 x +18 c_{1}}-\frac {1}{3} \\ \end{align*}
Verification of solutions
\[ y = \frac {\left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{9 x +9 c_{1}}-\frac {1}{3} \] Verified OK.
\[ y = -\frac {\left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{18 \left (x +c_{1} \right )}-\frac {i \sqrt {3}\, \left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{18 \left (x +c_{1} \right )}-\frac {1}{3} \] Verified OK.
\[ y = -\frac {\left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{18 \left (x +c_{1} \right )}+\frac {i \sqrt {3}\, \left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{18 x +18 c_{1}}-\frac {1}{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\left (3 y+1\right )^{4}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\left (3 y+1\right )^{4} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\left (3 y+1\right )^{4}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\left (3 y+1\right )^{4}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{9 \left (3 y+1\right )^{3}}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{9 \left (x +c_{1} \right )}-\frac {1}{3} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 118
dsolve(diff(y(x),x)=(3*y(x)+1)^4,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {3^{\frac {1}{3}} \left (-\left (c_{1} +x \right )^{2}\right )^{\frac {1}{3}}-3 c_{1} -3 x}{9 c_{1} +9 x} \\ y \left (x \right ) &= \frac {\left (-i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) \left (-\left (c_{1} +x \right )^{2}\right )^{\frac {1}{3}}-6 x -6 c_{1}}{18 c_{1} +18 x} \\ y \left (x \right ) &= \frac {\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) \left (-\left (c_{1} +x \right )^{2}\right )^{\frac {1}{3}}-6 x -6 c_{1}}{18 c_{1} +18 x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.928 (sec). Leaf size: 144
DSolve[y'[x]==(3*y[x]+1)^4,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {3 x+\sqrt [3]{3} \sqrt [3]{(x+c_1){}^2}+3 c_1}{9 (x+c_1)} \\ y(x)\to \frac {-6 x+\sqrt [3]{3} \left (1-i \sqrt {3}\right ) \sqrt [3]{(x+c_1){}^2}-6 c_1}{18 (x+c_1)} \\ y(x)\to \frac {-6 x+\sqrt [3]{3} \left (1+i \sqrt {3}\right ) \sqrt [3]{(x+c_1){}^2}-6 c_1}{18 (x+c_1)} \\ y(x)\to -\frac {1}{3} \\ \end{align*}