Internal problem ID [5735]
Internal file name [OUTPUT/4983_Sunday_June_05_2022_03_15_54_PM_7171702/index.tex]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 1. First order differential equations. Section 1.1 Separable equations problems.
page 7
Problem number: 22.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {y^{\prime }-\frac {\sqrt {y}}{\sqrt {x}}=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {\sqrt {y}}{\sqrt {x}} \end {align*}
Where \(f(x)=\frac {1}{\sqrt {x}}\) and \(g(y)=\sqrt {y}\). Integrating both sides gives \begin{align*} \frac {1}{\sqrt {y}} \,dy &= \frac {1}{\sqrt {x}} \,d x \\ \int { \frac {1}{\sqrt {y}} \,dy} &= \int {\frac {1}{\sqrt {x}} \,d x} \\ 2 \sqrt {y}&=2 \sqrt {x}+c_{1} \\ \end{align*} The solution is \[ 2 \sqrt {y}-2 \sqrt {x}-c_{1} = 0 \]
The solution(s) found are the following \begin{align*} \tag{1} 2 \sqrt {y}-2 \sqrt {x}-c_{1} &= 0 \\ \end{align*}
Verification of solutions
\[ 2 \sqrt {y}-2 \sqrt {x}-c_{1} = 0 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\frac {\sqrt {y}}{\sqrt {x}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\sqrt {y}}{\sqrt {x}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\sqrt {y}}=\frac {1}{\sqrt {x}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\sqrt {y}}d x =\int \frac {1}{\sqrt {x}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & 2 \sqrt {y}=2 \sqrt {x}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\sqrt {x}\, c_{1} +\frac {c_{1}^{2}}{4}+x \end {array} \]
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: 16
dsolve(diff(y(x),x)=sqrt(y(x))/sqrt(x),y(x), singsol=all)
\[ \sqrt {y \left (x \right )}-\sqrt {x}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.14 (sec). Leaf size: 26
DSolve[y'[x]==Sqrt[y[x]]/Sqrt[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{4} \left (2 \sqrt {x}+c_1\right ){}^2 \\ y(x)\to 0 \\ \end{align*}