Internal problem ID [12945]
Internal file name [OUTPUT/11598_Tuesday_November_07_2023_11_51_50_PM_63682322/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 1.5 page 71
Problem number: 12.
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]
\[ \boxed {y^{\prime }+y^{2}=0} \]
Integrating both sides gives \begin {align*} \int -\frac {1}{y^{2}}d y &= t +c_{1}\\ \frac {1}{y}&=t +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {1}{t +c_{1}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {1}{t +c_{1}} \\ \end{align*}
Verification of solutions
\[ y = \frac {1}{t +c_{1}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+y^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-y^{2} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{2}}=-1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{y^{2}}d t =\int \left (-1\right )d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{y}=-t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {1}{-t +c_{1}} \end {array} \]
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: 0.016 (sec). Leaf size: 9
dsolve(diff(y(t),t)=-y(t)^2,y(t), singsol=all)
\[ y \left (t \right ) = \frac {1}{t +c_{1}} \]
✓ Solution by Mathematica
Time used: 0.156 (sec). Leaf size: 18
DSolve[y'[t]==-y[t]^2,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{t-c_1} \\ y(t)\to 0 \\ \end{align*}