Internal problem ID [5730]
Internal file name [OUTPUT/4978_Sunday_June_05_2022_03_15_47_PM_86994772/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: 17.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {\frac {y^{\prime }}{\sqrt {y}}=-\frac {1}{\sqrt {x}}} \]
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.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\sqrt {y}}=-\frac {1}{\sqrt {x}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\sqrt {y}}d x =\int -\frac {1}{\sqrt {x}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & 2 \sqrt {y}=-2 \sqrt {x}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\sqrt {x}\, c_{1} +\frac {c_{1}^{2}}{4}+x \end {array} \]
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: 0.0 (sec). Leaf size: 14
dsolve(1/sqrt(x)+diff(y(x),x)/sqrt(y(x))=0,y(x), singsol=all)
\[ \sqrt {y \left (x \right )}+\sqrt {x}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.125 (sec). Leaf size: 21
DSolve[1/Sqrt[x]+y'[x]/Sqrt[y[x]]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{4} \left (-2 \sqrt {x}+c_1\right ){}^2 \]