Internal problem ID [2572]
Internal file name [OUTPUT/2064_Sunday_June_05_2022_02_46_55_AM_66233644/index.tex]
Book: Differential equations and linear algebra, Stephen W. Goode, second edition,
2000
Section: 1.6, page 50
Problem number: 14.
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]
\[ \boxed {y^{\prime }=-\frac {m}{x}+\ln \left (x \right )} \]
Integrating both sides gives \begin {align*} y &= \int { \frac {\ln \left (x \right ) x -m}{x}\,\mathop {\mathrm {d}x}}\\ &= -m \ln \left (x \right )+\ln \left (x \right ) x -x +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -m \ln \left (x \right )+\ln \left (x \right ) x -x +c_{1} \\ \end{align*}
Verification of solutions
\[ y = -m \ln \left (x \right )+\ln \left (x \right ) x -x +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {m}{x}+\ln \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \left (-\frac {m}{x}+\ln \left (x \right )\right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-m \ln \left (x \right )+\ln \left (x \right ) x -x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-m \ln \left (x \right )+\ln \left (x \right ) x -x +c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 17
dsolve(diff(y(x),x)+m/x=ln(x),y(x), singsol=all)
\[ y \left (x \right ) = \left (-m +x \right ) \ln \left (x \right )+c_{1} -x \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 19
DSolve[y'[x]+m/x==Log[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (x-m) \log (x)-x+c_1 \]