Internal problem ID [3663]
Internal file name [OUTPUT/3156_Sunday_June_05_2022_08_54_00_AM_71910197/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 15
Problem number: 409.
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]
\[ \boxed {X^{\frac {2}{3}} y^{\prime }-Y^{\frac {2}{3}}=0} \]
Integrating both sides gives \begin {align*} y &= \int { \frac {Y^{\frac {2}{3}}}{X^{\frac {2}{3}}}\,\mathop {\mathrm {d}x}}\\ &= \frac {x \,Y^{\frac {2}{3}}}{X^{\frac {2}{3}}}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x \,Y^{\frac {2}{3}}}{X^{\frac {2}{3}}}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {x \,Y^{\frac {2}{3}}}{X^{\frac {2}{3}}}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & X^{\frac {2}{3}} y^{\prime }-Y^{\frac {2}{3}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {Y^{\frac {2}{3}}}{X^{\frac {2}{3}}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {Y^{\frac {2}{3}}}{X^{\frac {2}{3}}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {x \,Y^{\frac {2}{3}}}{X^{\frac {2}{3}}}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {x \,Y^{\frac {2}{3}}+c_{1} X^{\frac {2}{3}}}{X^{\frac {2}{3}}} \end {array} \]
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: 0.016 (sec). Leaf size: 14
dsolve(X^(2/3)*diff(y(x),x) = Y^(2/3),y(x), singsol=all)
\[ y \left (x \right ) = \frac {Y^{\frac {2}{3}} x}{X^{\frac {2}{3}}}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 20
DSolve[X^(2/3) y'[x]== Y^(2/3),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x Y^{2/3}}{X^{2/3}}+c_1 \]