Internal problem ID [526]
Internal file name [OUTPUT/526_Sunday_June_05_2022_01_43_04_AM_90328302/index.tex]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Section 2.4. Page 76
Problem number: 15.
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]
\[ \boxed {y^{3}+y^{\prime }=0} \]
Integrating both sides gives \begin {align*} \int -\frac {1}{y^{3}}d y &= t +c_{1}\\ \frac {1}{2 y^{2}}&=t +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {1}{\sqrt {2 c_{1} +2 t}}\\ y_2&=-\frac {1}{\sqrt {2 c_{1} +2 t}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {1}{\sqrt {2 c_{1} +2 t}} \\ \tag{2} y &= -\frac {1}{\sqrt {2 c_{1} +2 t}} \\ \end{align*}
Verification of solutions
\[ y = \frac {1}{\sqrt {2 c_{1} +2 t}} \] Verified OK.
\[ y = -\frac {1}{\sqrt {2 c_{1} +2 t}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{3}+y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-y^{3} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{3}}=-1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{y^{3}}d t =\int \left (-1\right )d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{2 y^{2}}=-t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\frac {1}{\sqrt {-2 c_{1} +2 t}}, y=-\frac {1}{\sqrt {-2 c_{1} +2 t}}\right \} \end {array} \]
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: 0.0 (sec). Leaf size: 23
dsolve(y(t)^3+diff(y(t),t) = 0,y(t), singsol=all)
\begin{align*} y \left (t \right ) &= \frac {1}{\sqrt {2 t +c_{1}}} \\ y \left (t \right ) &= -\frac {1}{\sqrt {2 t +c_{1}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.124 (sec). Leaf size: 40
DSolve[y[t]^3+y'[t] == 0,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to -\frac {1}{\sqrt {2 t-2 c_1}} \\ y(t)\to \frac {1}{\sqrt {2 t-2 c_1}} \\ y(t)\to 0 \\ \end{align*}