Internal problem ID [12680]
Internal file name [OUTPUT/11333_Friday_November_03_2023_06_30_33_AM_84621573/index.tex]
Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 2. The Initial Value Problem. Exercises 2.3.2, page 63
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]
\[ \boxed {2 y y^{\prime }=1} \]
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.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 y y^{\prime }=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y y^{\prime }=\frac {1}{2} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y y^{\prime }d x =\int \frac {1}{2}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{2}}{2}=\frac {x}{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\sqrt {2 c_{1} +x}, y=-\sqrt {2 c_{1} +x}\right \} \end {array} \]
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: 0.0 (sec). Leaf size: 19
dsolve(2*y(x)*diff(y(x),x)=1,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: 0.019 (sec). Leaf size: 31
DSolve[2*y[x]*y'[x]==1,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*}