Internal problem ID [7054]
Internal file name [OUTPUT/6040_Sunday_June_05_2022_04_14_56_PM_58924344/index.tex]
Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 10.
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]
\[ \boxed {y^{\prime }=1+\frac {\sec \left (x \right )}{x}} \]
Integrating both sides gives \begin {align*} y = \int \frac {x +\sec \left (x \right )}{x}d x +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \int \frac {x +\sec \left (x \right )}{x}d x +c_{1} \\ \end{align*}
Verification of solutions
\[ y = \int \frac {x +\sec \left (x \right )}{x}d x +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=1+\frac {\sec \left (x \right )}{x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \left (1+\frac {\sec \left (x \right )}{x}\right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=x +\int \frac {2 \,{\mathrm e}^{\mathrm {I} x}}{\left (\left ({\mathrm e}^{\mathrm {I} x}\right )^{2}+1\right ) x}d x +c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 15
dsolve(diff(y(x),x) = 1+sec(x)/x,y(x), singsol=all)
\[ y \left (x \right ) = \int \frac {\sec \left (x \right )}{x}d x +x +c_{1} \]
✓ Solution by Mathematica
Time used: 0.833 (sec). Leaf size: 25
DSolve[y'[x] == 1+Sec[x]/x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \int _1^x\left (\frac {\sec (K[1])}{K[1]}+1\right )dK[1]+c_1 \]