problem 903

Internal problem ID [9236]
Internal ﬁle name [OUTPUT/8172_Monday_June_06_2022_02_02_05_AM_45371114/index.tex]

Book: Diﬀerential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear ﬁrst order
Problem number: 903.
ODE order: 1.
ODE degree: 1.

\[ \boxed {y^{\prime }-\frac {\sin \left (\frac {y}{x}\right ) \left (y+2 x^{2} \sin \left (\frac {y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )\right )}{2 \sin \left (\frac {y}{2 x}\right ) x \cos \left (\frac {y}{2 x}\right )}=0} \]

2.326.1 Solving as homogeneousTypeD2 ode

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*} 2 \sin \left (u \left (x \right )\right ) \sin \left (\frac {u \left (x \right )}{2}\right ) \cos \left (\frac {u \left (x \right )}{2}\right ) x^{2}-2 \left (u^{\prime }\left (x \right ) x +u \left (x \right )\right ) \sin \left (\frac {u \left (x \right )}{2}\right ) x \cos \left (\frac {u \left (x \right )}{2}\right )+\sin \left (u \left (x \right )\right ) u \left (x \right ) x = 0 \end {align*}

In canonical form the ODE is \begin {align*} u' &= F(x,u)\\ &= f( x) g(u)\\ &= \sin \left (u \right ) \end {align*}

Where \(f(x)=1\) and \(g(u)=\sin \left (u \right )\). Integrating both sides gives \begin{align*} \frac {1}{\sin \left (u \right )} \,du &= 1 \,d x \\ \int { \frac {1}{\sin \left (u \right )} \,du} &= \int {1 \,d x} \\ \ln \left (\tan \left (\frac {u}{2}\right )\right )&=x +c_{2} \\ \end{align*} Raising both side to exponential gives \begin {align*} \tan \left (\frac {u}{2}\right ) &= {\mathrm e}^{x +c_{2}} \end {align*}

Which simpliﬁes to \begin {align*} \tan \left (\frac {u}{2}\right ) &= c_{3} {\mathrm e}^{x} \end {align*}

Therefore the solution \(y\) is \begin {align*} y&=x u\\ &=2 x \arctan \left (c_{3} {\mathrm e}^{x +c_{2}}\right ) \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= 2 x \arctan \left (c_{3} {\mathrm e}^{x +c_{2}}\right ) \\ \end{align*}

Veriﬁcation of solutions

\[ y = 2 x \arctan \left (c_{3} {\mathrm e}^{x +c_{2}}\right ) \] Veriﬁed OK.

2.326.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 \sin \left (\frac {y}{x}\right ) \sin \left (\frac {y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right ) x^{2}-2 y^{\prime } \sin \left (\frac {y}{2 x}\right ) x \cos \left (\frac {y}{2 x}\right )+\sin \left (\frac {y}{x}\right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-2 \sin \left (\frac {y}{x}\right ) \sin \left (\frac {y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right ) x^{2}-\sin \left (\frac {y}{x}\right ) y}{2 \sin \left (\frac {y}{2 x}\right ) x \cos \left (\frac {y}{2 x}\right )} \end {array} \]

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`

\[ y \left (x \right ) = \arctan \left (\frac {2 c_{1} {\mathrm e}^{x}}{{\mathrm e}^{2 x} c_{1}^{2}+1}, \frac {-{\mathrm e}^{2 x} c_{1}^{2}+1}{{\mathrm e}^{2 x} c_{1}^{2}+1}\right ) x \]

\begin{align*} y(x)\to -x \arccos (-\tanh (x+c_1)) \\ y(x)\to x \arccos (-\tanh (x+c_1)) \\ y(x)\to 0 \\ y(x)\to -\pi x \\ y(x)\to \pi x \\ \end{align*}

2.326 problem 903

2.326.1 Solving as homogeneousTypeD2 ode

2.326.2 Maple step by step solution