Internal problem ID [12917]
Internal file name [OUTPUT/11570_Tuesday_November_07_2023_11_27_13_PM_2375291/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: 16 (iv).
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]
\[ \boxed {y^{\prime }=-t^{2}+2} \]
Integrating both sides gives \begin {align*} y &= \int { -t^{2}+2\,\mathop {\mathrm {d}t}}\\ &= -\frac {t \left (t^{2}-6\right )}{3}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {t \left (t^{2}-6\right )}{3}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = -\frac {t \left (t^{2}-6\right )}{3}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=-t^{2}+2 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int y^{\prime }d t =\int \left (-t^{2}+2\right )d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\frac {1}{3} t^{3}+2 t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {1}{3} t^{3}+2 t +c_{1} \end {array} \]
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: 14
dsolve(diff(y(t),t)=2-t^2,y(t), singsol=all)
\[ y \left (t \right ) = -\frac {1}{3} t^{3}+2 t +c_{1} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 18
DSolve[y'[t]==2-t^2,y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to -\frac {t^3}{3}+2 t+c_1 \]