Internal problem ID [5881]
Internal file name [OUTPUT/5129_Sunday_June_05_2022_03_25_37_PM_22330115/index.tex]
Book: THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS.
K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section: Chapter 3. Ordinary Differential Equations. Section 3.6 Summary and Problems. Page
218
Problem number: Problem 3.8.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "homogeneousTypeD2"
Maple gives the following as the ode type
[[_homogeneous, `class D`]]
\[ \boxed {\frac {1}{y}+\sec \left (\frac {y}{x}\right )-\frac {x y^{\prime }}{y^{2}}=0} \]
Using the change of variables \(y = u \left (x \right ) x\) on the above ode results in new ode in \(u \left (x \right )\) \begin {align*} \sec \left (u \left (x \right )\right ) u \left (x \right )^{2} x^{2}-\left (u^{\prime }\left (x \right ) x +u \left (x \right )\right ) x +u \left (x \right ) x = 0 \end {align*}
Integrating both sides gives \begin {align*} \int \frac {1}{\sec \left (u \right ) u^{2}}d u &= \int {dx}\\ \int _{}^{u \left (x \right )}\frac {1}{\sec \left (\textit {\_a} \right ) \textit {\_a}^{2}}d \textit {\_a}&= x +c_{2} \end {align*}
Replacing \(u(x)\) in the above solution by \(\frac {y}{x}\) results in the solution for \(y\) in implicit form \begin {align*} \int _{}^{\frac {y}{x}}\frac {1}{\sec \left (\textit {\_a} \right ) \textit {\_a}^{2}}d \textit {\_a} = x +c_{2}\\ \int _{}^{\frac {y}{x}}\frac {\cos \left (\textit {\_a} \right )}{\textit {\_a}^{2}}d \textit {\_a} = x +c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{\frac {y}{x}}\frac {\cos \left (\textit {\_a} \right )}{\textit {\_a}^{2}}d \textit {\_a} &= x +c_{2} \\ \end{align*}
Verification of solutions
\[ \int _{}^{\frac {y}{x}}\frac {\cos \left (\textit {\_a} \right )}{\textit {\_a}^{2}}d \textit {\_a} = x +c_{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \sec \left (\frac {y}{x}\right ) y^{2}-y^{\prime } x +y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\sec \left (\frac {y}{x}\right ) y^{2}+y}{x} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying homogeneous D <- homogeneous successful`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 20
dsolve((1/y(x)+sec(y(x)/x))-x/y(x)^2*diff(y(x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \operatorname {RootOf}\left (\textit {\_Z} \,\operatorname {Si}\left (\textit {\_Z} \right )+\textit {\_Z} c_{1} +\textit {\_Z} x +\cos \left (\textit {\_Z} \right )\right ) x \]
✓ Solution by Mathematica
Time used: 0.145 (sec). Leaf size: 32
DSolve[(1/y[x]+Sec[y[x]/x])-x/y[x]^2*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\text {Si}\left (\frac {y(x)}{x}\right )-\frac {x \cos \left (\frac {y(x)}{x}\right )}{y(x)}=x+c_1,y(x)\right ] \]