Internal problem ID [14340]
Internal file name [OUTPUT/14021_Monday_March_11_2024_01_27_18_AM_23470582/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: 59 (i).
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]
\[ \boxed {2 y+\left (2 t +2 y\right ) y^{\prime }=-2 t} \]
Integrating both sides gives \begin {align*} y &= \int { -1\,\mathop {\mathrm {d}t}}\\ &= -t +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -t +c_{1} \\ \end{align*}
Verification of solutions
\[ y = -t +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 y+\left (2 t +2 y\right ) y^{\prime }=-2 t \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & 2 t y^{\prime }+2 y^{\prime } y+2 y=-2 t \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \left (2 t y^{\prime }+2 y^{\prime } y+2 y\right )d t =\int -2 t d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y^{2}+2 y t =-t^{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=-t -\sqrt {c_{1}}, y=-t +\sqrt {c_{1}}\right \} \end {array} \]
Maple trace
`Classification methods on request Methods to be used are: [exact] ---------------------------- * Tackling ODE using method: exact --- Trying classification methods --- trying exact <- exact successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 25
dsolve((2*t+2*y(t))+(2*t+2*y(t))*diff(y(t),t)=0,y(t), singsol=all)
\begin{align*} y \left (t \right ) &= -t \\ y \left (t \right ) &= -c_{1} -t \\ y \left (t \right ) &= -t +c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 18
DSolve[(2*t+2*y[t])+(2*t+2*y[t])*y'[t]==0,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to -t \\ y(t)\to -t+c_1 \\ \end{align*}