Internal problem ID [12899]
Internal file name [OUTPUT/11552_Monday_November_06_2023_01_33_19_PM_34935870/index.tex]
Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th
edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 1. First-Order Differential Equations. Exercises section 1.3 page 47
Problem number: 1.
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]
\[ \boxed {y^{\prime }=t^{2}+t} \]
Integrating both sides gives \begin {align*} y &= \int { t^{2}+t\,\mathop {\mathrm {d}t}}\\ &= \frac {t^{2} \left (2 t +3\right )}{6}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {t^{2} \left (2 t +3\right )}{6}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {t^{2} \left (2 t +3\right )}{6}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=t^{2}+t \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int y^{\prime }d t =\int \left (t^{2}+t \right )d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {1}{3} t^{3}+\frac {1}{2} t^{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {1}{3} t^{3}+\frac {1}{2} t^{2}+c_{1} \end {array} \]
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: 16
dsolve(diff(y(t),t)=t^2+t,y(t), singsol=all)
\[ y \left (t \right ) = \frac {1}{3} t^{3}+\frac {1}{2} t^{2}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 22
DSolve[y'[t]==t^2+t,y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {t^3}{3}+\frac {t^2}{2}+c_1 \]