Internal problem ID [12578]
Internal file name [OUTPUT/11231_Thursday_October_19_2023_04_43_31_PM_58456927/index.tex]
Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 1. Introduction. Exercises page 14
Problem number: 20.
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]
\[ \boxed {y^{\prime }+\frac {1}{2 y}=0} \]
Integrating both sides gives \begin {align*} \int -2 y d y &= x +c_{1}\\ -y^{2}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\sqrt {-x -c_{1}}\\ y_2&=-\sqrt {-x -c_{1}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \sqrt {-x -c_{1}} \\ \tag{2} y &= -\sqrt {-x -c_{1}} \\ \end{align*}
Verification of solutions
\[ y = \sqrt {-x -c_{1}} \] Verified OK.
\[ y = -\sqrt {-x -c_{1}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+\frac {1}{2 y}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {1}{2 y} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y y^{\prime }=-\frac {1}{2} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y y^{\prime }d x =\int -\frac {1}{2}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{2}}{2}=-\frac {x}{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\sqrt {2 c_{1} -x}, y=-\sqrt {2 c_{1} -x}\right \} \end {array} \]
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: 23
dsolve(diff(y(x),x)+1/(2*y(x))=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \sqrt {c_{1} -x} \\ y \left (x \right ) &= -\sqrt {c_{1} -x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.113 (sec). Leaf size: 35
DSolve[y'[x]+1/(2*y[x])==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {-x+2 c_1} \\ y(x)\to \sqrt {-x+2 c_1} \\ \end{align*}