Internal problem ID [884]
Internal file name [OUTPUT/884_Sunday_June_05_2022_01_53_10_AM_42109854/index.tex]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 1, Introduction. Section 1.2 Page 14
Problem number: 8(a).
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]
\[ \boxed {y^{\prime }-a y^{\frac {a -1}{a}}=0} \]
Integrating both sides gives \begin{align*} \int \frac {y^{-\frac {a -1}{a}}}{a}d y &= \int d x \\ y \,y^{-\frac {a -1}{a}}&=x +c_{1} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y \,y^{-\frac {a -1}{a}} &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ y \,y^{-\frac {a -1}{a}} = x +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-a y^{\frac {a -1}{a}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=a y^{\frac {a -1}{a}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{\frac {a -1}{a}}}=a \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{\frac {a -1}{a}}}d x =\int a d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{-\frac {a -1}{a}+1}}{-\frac {a -1}{a}+1}=x a +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{\ln \left (\frac {x a +c_{1}}{a}\right ) a} \end {array} \]
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: 9
dsolve(diff(y(x),x) = a*y(x)^( (a-1)/a),y(x), singsol=all)
\[ y \left (x \right ) = \left (c_{1} +x \right )^{a} \]
✓ Solution by Mathematica
Time used: 0.843 (sec). Leaf size: 28
DSolve[y'[x] ==a*y[x]^( (a-1)/a),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (x+\frac {c_1}{a}\right ){}^a \\ y(x)\to 0^{\frac {a}{a-1}} \\ \end{align*}