4.17 problem 106

Internal problem ID [3364]
Internal file name [OUTPUT/2856_Sunday_June_05_2022_08_41_56_AM_38490149/index.tex]

Book: Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 4
Problem number: 106.
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]

\[ \boxed {y^{\prime }-\sqrt {X Y}=0} \]

4.17.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} y &= \int { \sqrt {X Y}\,\mathop {\mathrm {d}x}}\\ &= x \sqrt {X Y}+c_{1} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= x \sqrt {X Y}+c_{1} \\ \end{align*}

Verification of solutions

\[ y = x \sqrt {X Y}+c_{1} \] Verified OK.

4.17.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\sqrt {X Y}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime }=\sqrt {X Y} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \sqrt {X Y}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=x \sqrt {X Y}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=x \sqrt {X Y}+c_{1} \end {array} \]

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: 13

dsolve(diff(y(x),x) = sqrt(X*Y),y(x), singsol=all)
 

\[ y \left (x \right ) = \sqrt {X Y}\, x +c_{1} \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 17

DSolve[y'[x]==Sqrt[ X Y],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to x \sqrt {X Y}+c_1 \]