Internal problem ID [6143]
Internal file name [OUTPUT/5391_Sunday_June_05_2022_03_36_04_PM_1499382/index.tex]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.3 SEPARABLE EQUATIONS. Page
12
Problem number: 1(e).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {y \ln \left (y\right )-x y^{\prime }=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {y \ln \left (y \right )}{x} \end {align*}
Where \(f(x)=\frac {1}{x}\) and \(g(y)=\ln \left (y \right ) y\). Integrating both sides gives \begin{align*} \frac {1}{\ln \left (y \right ) y} \,dy &= \frac {1}{x} \,d x \\ \int { \frac {1}{\ln \left (y \right ) y} \,dy} &= \int {\frac {1}{x} \,d x} \\ \ln \left (\ln \left (y \right )\right )&=\ln \left (x \right )+c_{1} \\ \end{align*} Raising both side to exponential gives \begin {align*} \ln \left (y \right ) &= {\mathrm e}^{\ln \left (x \right )+c_{1}} \end {align*}
Which simplifies to \begin {align*} \ln \left (y \right ) &= c_{2} x \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{c_{2} {\mathrm e}^{c_{1}} x} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{c_{2} {\mathrm e}^{c_{1}} x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \ln \left (y\right )-x y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y \ln \left (y\right )}{x} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y \ln \left (y\right )}=\frac {1}{x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y \ln \left (y\right )}d x =\int \frac {1}{x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (\ln \left (y\right )\right )=\ln \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{{\mathrm e}^{c_{1}} x} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 8
dsolve(y(x)*ln(y(x))-x*diff(y(x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{c_{1} x} \]
✓ Solution by Mathematica
Time used: 0.172 (sec). Leaf size: 18
DSolve[y[x]*Log[y[x]]-x*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{e^{c_1} x} \\ y(x)\to 1 \\ \end{align*}