2.6 problem 1(f)

Internal problem ID [6144]
Internal file name [OUTPUT/5392_Sunday_June_05_2022_03_36_06_PM_57599098/index.tex]

Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.3 SEPARABLE EQUATIONS. Page 12
Problem number: 1(f).
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program :

Maple gives the following as the ode type

[_separable]

\[ \boxed {x y^{\prime }-\left (-4 x^{2}+1\right ) \tan \left (y\right )=0} \]

2.6.1 Solving as separable ode

In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -\frac {\tan \left (y \right ) \left (4 x^{2}-1\right )}{x} \end {align*}

Where \(f(x)=-\frac {4 x^{2}-1}{x}\) and \(g(y)=\tan \left (y \right )\). Integrating both sides gives \begin{align*} \frac {1}{\tan \left (y \right )} \,dy &= -\frac {4 x^{2}-1}{x} \,d x \\ \int { \frac {1}{\tan \left (y \right )} \,dy} &= \int {-\frac {4 x^{2}-1}{x} \,d x} \\ \ln \left (\sin \left (y \right )\right )&=-2 x^{2}+\ln \left (x \right )+c_{1} \\ \end{align*} Raising both side to exponential gives \begin {align*} \sin \left (y \right ) &= {\mathrm e}^{-2 x^{2}+\ln \left (x \right )+c_{1}} \end {align*}

Which simplifies to \begin {align*} \sin \left (y \right ) &= c_{2} {\mathrm e}^{-2 x^{2}+\ln \left (x \right )} \end {align*}

Summary

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

Figure 54: Slope field plot

Verification of solutions

\[ y = \arcsin \left (c_{2} {\mathrm e}^{-2 x^{2}+c_{1}} x \right ) \] Verified OK.

2.6.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y^{\prime }-\left (-4 x^{2}+1\right ) \tan \left (y\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\left (-4 x^{2}+1\right ) \tan \left (y\right )}{x} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\tan \left (y\right )}=\frac {-4 x^{2}+1}{x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\tan \left (y\right )}d x =\int \frac {-4 x^{2}+1}{x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (\sin \left (y\right )\right )=-2 x^{2}+\ln \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\arcsin \left (\frac {x}{{\mathrm e}^{2 x^{2}-c_{1}}}\right ) \end {array} \]

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
<- separable successful`
 

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 16

dsolve(x*diff(y(x),x)=(1-4*x^2)*tan(y(x)),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 53.453 (sec). Leaf size: 23

DSolve[x*y'[x]==(1-4*x^2)*Tan[y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \arcsin \left (x e^{-2 x^2+c_1}\right ) \\ y(x)\to 0 \\ \end{align*}