2.4.7 problem 7

Internal problem ID [18234]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter IV. Methods of solution: First order equations. section 31. Problems at page 85
Problem number : 7
Date solved : Monday, December 23, 2024 at 09:17:15 PM
CAS classification : [_Bernoulli]

Solve

\begin{align*} y-\cos \left (x \right ) y^{\prime }&=y^{2} \cos \left (x \right ) \left (1-\sin \left (x \right )\right ) \end{align*}

Solved as first order Bernoulli ode

Time used: 0.448 (sec)

In canonical form, the ODE is

\begin{align*} y' &= F(x,y)\\ &= \frac {y \left (y \cos \left (x \right ) \sin \left (x \right )-y \cos \left (x \right )+1\right )}{\cos \left (x \right )} \end{align*}

This is a Bernoulli ODE.

\[ y' = \left (\frac {1}{\cos \left (x \right )}\right ) y + \left (\frac {\cos \left (x \right ) \sin \left (x \right )-\cos \left (x \right )}{\cos \left (x \right )}\right )y^{2} \tag {1} \]

The standard Bernoulli ODE has the form

\[ y' = f_0(x)y+f_1(x)y^n \tag {2} \]

Comparing this to (1) shows that

\begin{align*} f_0 &=\frac {1}{\cos \left (x \right )}\\ f_1 &=\frac {\cos \left (x \right ) \sin \left (x \right )-\cos \left (x \right )}{\cos \left (x \right )} \end{align*}

The first step is to divide the above equation by \(y^n \) which gives

\[ \frac {y'}{y^n} = f_0(x) y^{1-n} +f_1(x) \tag {3} \]

The next step is use the substitution \(v = y^{1-n}\) in equation (3) which generates a new ODE in \(v \left (x \right )\) which will be linear and can be easily solved using an integrating factor. Backsubstitution then gives the solution \(y(x)\) which is what we want.

This method is now applied to the ODE at hand. Comparing the ODE (1) With (2) Shows that

\begin{align*} f_0(x)&=\frac {1}{\cos \left (x \right )}\\ f_1(x)&=\frac {\cos \left (x \right ) \sin \left (x \right )-\cos \left (x \right )}{\cos \left (x \right )}\\ n &=2 \end{align*}

Dividing both sides of ODE (1) by \(y^n=y^{2}\) gives

\begin{align*} y'\frac {1}{y^{2}} &= \frac {1}{\cos \left (x \right ) y} +\frac {\cos \left (x \right ) \sin \left (x \right )-\cos \left (x \right )}{\cos \left (x \right )} \tag {4} \end{align*}

Let

\begin{align*} v &= y^{1-n} \\ &= \frac {1}{y} \tag {5} \end{align*}

Taking derivative of equation (5) w.r.t \(x\) gives

\begin{align*} v' &= -\frac {1}{y^{2}}y' \tag {6} \end{align*}

Substituting equations (5) and (6) into equation (4) gives

\begin{align*} -v^{\prime }\left (x \right )&= \frac {v \left (x \right )}{\cos \left (x \right )}+\frac {\cos \left (x \right ) \sin \left (x \right )-\cos \left (x \right )}{\cos \left (x \right )}\\ v' &= -\frac {v}{\cos \left (x \right )}-\frac {\cos \left (x \right ) \sin \left (x \right )-\cos \left (x \right )}{\cos \left (x \right )} \tag {7} \end{align*}

The above now is a linear ODE in \(v \left (x \right )\) which is now solved.

In canonical form a linear first order is

\begin{align*} v^{\prime }\left (x \right ) + q(x)v \left (x \right ) &= p(x) \end{align*}

Comparing the above to the given ode shows that

\begin{align*} q(x) &=\sec \left (x \right )\\ p(x) &=1-\sin \left (x \right ) \end{align*}

The integrating factor \(\mu \) is

\begin{align*} \mu &= e^{\int {q\,dx}}\\ &= {\mathrm e}^{\int \sec \left (x \right )d x}\\ &= \sec \left (x \right )+\tan \left (x \right ) \end{align*}

The ode becomes

\begin{align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu v\right ) &= \mu p \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu v\right ) &= \left (\mu \right ) \left (1-\sin \left (x \right )\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (v \left (\sec \left (x \right )+\tan \left (x \right )\right )\right ) &= \left (\sec \left (x \right )+\tan \left (x \right )\right ) \left (1-\sin \left (x \right )\right ) \\ \mathrm {d} \left (v \left (\sec \left (x \right )+\tan \left (x \right )\right )\right ) &= \left (\left (1-\sin \left (x \right )\right ) \left (\sec \left (x \right )+\tan \left (x \right )\right )\right )\, \mathrm {d} x \\ \end{align*}

Integrating gives

\begin{align*} v \left (\sec \left (x \right )+\tan \left (x \right )\right )&= \int {\left (1-\sin \left (x \right )\right ) \left (\sec \left (x \right )+\tan \left (x \right )\right ) \,dx} \\ &=\sin \left (x \right ) + c_1 \end{align*}

Dividing throughout by the integrating factor \(\sec \left (x \right )+\tan \left (x \right )\) gives the final solution

\[ v \left (x \right ) = \frac {\left (\sin \left (x \right )+c_1 \right ) \left (\cos \left (x \right )-\sin \left (x \right )+1\right )}{\cos \left (x \right )+\sin \left (x \right )+1} \]

The substitution \(v = y^{1-n}\) is now used to convert the above solution back to \(y\) which results in

\[ \frac {1}{y} = \frac {\left (\sin \left (x \right )+c_1 \right ) \left (\cos \left (x \right )-\sin \left (x \right )+1\right )}{\cos \left (x \right )+\sin \left (x \right )+1} \]

Solving for \(y\) gives

\begin{align*} y &= \frac {\cos \left (x \right )+\sin \left (x \right )+1}{\cos \left (x \right ) \sin \left (x \right )+\cos \left (x \right ) c_1 -\sin \left (x \right )^{2}-c_1 \sin \left (x \right )+\sin \left (x \right )+c_1} \\ \end{align*}

Figure 2.52: Slope field plot
\(y-\cos \left (x \right ) y^{\prime } = y^{2} \cos \left (x \right ) \left (1-\sin \left (x \right )\right )\)

Summary of solutions found

\begin{align*} y &= \frac {\cos \left (x \right )+\sin \left (x \right )+1}{\cos \left (x \right ) \sin \left (x \right )+\cos \left (x \right ) c_1 -\sin \left (x \right )^{2}-c_1 \sin \left (x \right )+\sin \left (x \right )+c_1} \\ \end{align*}

Maple step by step solution

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

Maple trace

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

Maple dsolve solution

Solving time : 0.003 (sec)
Leaf size : 27

dsolve(y(x)-cos(x)*diff(y(x),x) = y(x)^2*cos(x)*(1-sin(x)), 
       y(x),singsol=all)
 

\[ y = \frac {\cos \left (x \right )+\sin \left (x \right )+1}{\left (\sin \left (x \right )+c_1 \right ) \left (\cos \left (x \right )-\sin \left (x \right )+1\right )} \]

Mathematica DSolve solution

Solving time : 0.372 (sec)
Leaf size : 41

DSolve[{y[x]-Cos[x]*D[y[x],x]==y[x]^2*Cos[x]*(1-Sin[x]),{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {e^{2 \text {arctanh}\left (\tan \left (\frac {x}{2}\right )\right )}}{\cos (x) e^{2 \text {arctanh}\left (\tan \left (\frac {x}{2}\right )\right )}+c_1} \\ y(x)\to 0 \\ \end{align*}