Internal problem ID [3659]
Internal file name [OUTPUT/3152_Sunday_June_05_2022_08_53_52_AM_12371859/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 14
Problem number: 405.
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]
\[ \boxed {y^{\prime } \sqrt {X}+\sqrt {Y}=0} \]
Integrating both sides gives \begin {align*} y &= \int { -\frac {\sqrt {Y}}{\sqrt {X}}\,\mathop {\mathrm {d}x}}\\ &= -\frac {x \sqrt {Y}}{\sqrt {X}}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {x \sqrt {Y}}{\sqrt {X}}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = -\frac {x \sqrt {Y}}{\sqrt {X}}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } \sqrt {X}+\sqrt {Y}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {\sqrt {Y}}{\sqrt {X}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -\frac {\sqrt {Y}}{\sqrt {X}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\frac {x \sqrt {Y}}{\sqrt {X}}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {c_{1} \sqrt {X}-x \sqrt {Y}}{\sqrt {X}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 15
dsolve(diff(y(x),x)*sqrt(X)+sqrt(Y) = 0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\sqrt {Y}\, x}{\sqrt {X}}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 21
DSolve[y'[x] Sqrt[X]+Sqrt[Y]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {x \sqrt {Y}}{\sqrt {X}}+c_1 \]