Internal problem ID [2668]
Internal file name [OUTPUT/2160_Sunday_June_05_2022_02_50_57_AM_39886309/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 1.8, Change of Variables. page
79
Problem number: Problem 12.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[[_homogeneous, `class A`], _dAlembert]
\[ \boxed {\sin \left (\frac {y}{x}\right ) \left (y^{\prime } x -y\right )-x \cos \left (\frac {y}{x}\right )=0} \]
In canonical form, the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {x \cos \left (\frac {y}{x}\right )+\sin \left (\frac {y}{x}\right ) y}{\sin \left (\frac {y}{x}\right ) x}\tag {1} \end {align*}
An ode of the form \(y' = \frac {M(x,y)}{N(x,y)}\) is called homogeneous if the functions \(M(x,y)\) and \(N(x,y)\) are both homogeneous functions and of the same order. Recall that a function \(f(x,y)\) is homogeneous of order \(n\) if \[ f(t^n x, t^n y)= t^n f(x,y) \] In this case, it can be seen that both \(M=x \cos \left (\frac {y}{x}\right )+\sin \left (\frac {y}{x}\right ) y\) and \(N=\sin \left (\frac {y}{x}\right ) x\) are both homogeneous and of the same order \(n=1\). Therefore this is a homogeneous ode. Since this ode is homogeneous, it is converted to separable ODE using the substitution \(u=\frac {y}{x}\), or \(y=ux\). Hence \[ \frac { \mathop {\mathrm {d}y}}{\mathop {\mathrm {d}x}}= \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}x}}x + u \] Applying the transformation \(y=ux\) to the above ODE in (1) gives \begin {align*} \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}x}}x + u &= \frac {\cos \left (u \right )}{\sin \left (u \right )}+u\\ \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}x}} &= \frac {\cos \left (u \left (x \right )\right )}{\sin \left (u \left (x \right )\right ) x} \end {align*}
Or \[ u^{\prime }\left (x \right )-\frac {\cos \left (u \left (x \right )\right )}{\sin \left (u \left (x \right )\right ) x} = 0 \] Or \[ u^{\prime }\left (x \right ) \sin \left (u \left (x \right )\right ) x -\cos \left (u \left (x \right )\right ) = 0 \] Which is now solved as separable in \(u \left (x \right )\). Which is now solved in \(u \left (x \right )\). In canonical form the ODE is \begin {align*} u' &= F(x,u)\\ &= f( x) g(u)\\ &= \frac {\cot \left (u \right )}{x} \end {align*}
Where \(f(x)=\frac {1}{x}\) and \(g(u)=\cot \left (u \right )\). Integrating both sides gives \begin{align*} \frac {1}{\cot \left (u \right )} \,du &= \frac {1}{x} \,d x \\ \int { \frac {1}{\cot \left (u \right )} \,du} &= \int {\frac {1}{x} \,d x} \\ -\ln \left (\cos \left (u \right )\right )&=\ln \left (x \right )+c_{2} \\ \end{align*} Raising both side to exponential gives \begin {align*} \frac {1}{\cos \left (u \right )} &= {\mathrm e}^{\ln \left (x \right )+c_{2}} \end {align*}
Which simplifies to \begin {align*} \sec \left (u \right ) &= c_{3} x \end {align*}
Now \(u\) in the above solution is replaced back by \(y\) using \(u=\frac {y}{x}\) which results in the solution \[ y = x \,\operatorname {arcsec}\left (c_{3} {\mathrm e}^{c_{2}} x \right ) \]
The solution(s) found are the following \begin{align*} \tag{1} y &= x \,\operatorname {arcsec}\left (c_{3} {\mathrm e}^{c_{2}} x \right ) \\ \end{align*}
Verification of solutions
\[ y = x \,\operatorname {arcsec}\left (c_{3} {\mathrm e}^{c_{2}} x \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \sin \left (\frac {y}{x}\right ) \left (y^{\prime } x -y\right )-x \cos \left (\frac {y}{x}\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x \cos \left (\frac {y}{x}\right )+\sin \left (\frac {y}{x}\right ) y}{\sin \left (\frac {y}{x}\right ) x} \end {array} \]
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: 0.016 (sec). Leaf size: 14
dsolve(sin(y(x)/x)*(x*diff(y(x),x)-y(x))=x*cos(y(x)/x),y(x), singsol=all)
\[ y \left (x \right ) = x \arccos \left (\frac {1}{c_{1} x}\right ) \]
✓ Solution by Mathematica
Time used: 25.367 (sec). Leaf size: 56
DSolve[Sin[y[x]/x]*(x*y'[x]-y[x])==x*Cos[y[x]/x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -x \arccos \left (\frac {e^{-c_1}}{x}\right ) \\ y(x)\to x \arccos \left (\frac {e^{-c_1}}{x}\right ) \\ y(x)\to -\frac {\pi x}{2} \\ y(x)\to \frac {\pi x}{2} \\ \end{align*}