Internal problem ID [2625]
Internal file name [OUTPUT/2117_Sunday_June_05_2022_02_49_09_AM_36847014/index.tex]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth
edition, 2015
Section: Chapter 1, First-Order Differential Equations. Section 1.4, Separable Differential
Equations. page 43
Problem number: Problem 4.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {y^{\prime }-\frac {y}{\ln \left (x \right ) x}=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {y}{\ln \left (x \right ) x} \end {align*}
Where \(f(x)=\frac {1}{\ln \left (x \right ) x}\) and \(g(y)=y\). Integrating both sides gives \begin {align*} \frac {1}{y} \,dy &= \frac {1}{\ln \left (x \right ) x} \,d x\\ \int { \frac {1}{y} \,dy} &= \int {\frac {1}{\ln \left (x \right ) x} \,d x}\\ \ln \left (y \right )&=\ln \left (\ln \left (x \right )\right )+c_{1}\\ y&={\mathrm e}^{\ln \left (\ln \left (x \right )\right )+c_{1}}\\ &=c_{1} \ln \left (x \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \ln \left (x \right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \ln \left (x \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\frac {y}{\ln \left (x \right ) x}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y}{\ln \left (x \right ) x} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=\frac {1}{\ln \left (x \right ) x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int \frac {1}{\ln \left (x \right ) x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=\ln \left (\ln \left (x \right )\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{c_{1}} \ln \left (x \right ) \end {array} \]
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: 8
dsolve(diff(y(x),x)=y(x)/(x*ln(x)),y(x), singsol=all)
\[ y \left (x \right ) = \ln \left (x \right ) c_{1} \]
✓ Solution by Mathematica
Time used: 0.026 (sec). Leaf size: 15
DSolve[y'[x]==y[x]/(x*Log[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \log (x) \\ y(x)\to 0 \\ \end{align*}