Internal problem ID [14298]
Internal file name [OUTPUT/13979_Saturday_March_09_2024_04_42_33_PM_45166154/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: 11.
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]
\[ \boxed {-y^{\prime }=-3 t^{2}} \]
Integrating both sides gives \begin {align*} y &= \int { 3 t^{2}\,\mathop {\mathrm {d}t}}\\ &= t^{3}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= t^{3}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = t^{3}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y^{\prime }=-3 t^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime }=3 t^{2} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int y^{\prime }d t =\int 3 t^{2}d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=t^{3}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=t^{3}+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 9
dsolve(3*t^2-diff(y(t),t)=0,y(t), singsol=all)
\[ y \left (t \right ) = t^{3}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 11
DSolve[3*t^2-y'[t]==0,y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to t^3+c_1 \]