1.31 problem 31

Internal problem ID [6394]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.1 Separable equations problems. page 7
Problem number : 31
Date solved : Thursday, October 17, 2024 at 10:07:04 AM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

Solve

\begin{align*} y^{\prime }+\sin \left (x +y\right )^{2}&=0 \end{align*}

1.31.1 Solved using Lie symmetry for first order ode

Time used: 0.767 (sec)

Writing the ode as

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

The condition of Lie symmetry is the linearized PDE given by

\begin{align*} \eta _{x}+\omega \left ( \eta _{y}-\xi _{x}\right ) -\omega ^{2}\xi _{y}-\omega _{x}\xi -\omega _{y}\eta =0\tag {A} \end{align*}

To determine \(\xi ,\eta \) then (A) is solved using ansatz. Making bivariate polynomials of degree 1 to use as anstaz gives

\begin{align*} \tag{1E} \xi &= x a_{2}+y a_{3}+a_{1} \\ \tag{2E} \eta &= x b_{2}+y b_{3}+b_{1} \\ \end{align*}

Where the unknown coefficients are

\[ \{a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}\} \]

Substituting equations (1E,2E) and \(\omega \) into (A) gives

\begin{equation} \tag{5E} b_{2}-\sin \left (x +y \right )^{2} \left (b_{3}-a_{2}\right )-\sin \left (x +y \right )^{4} a_{3}+2 \sin \left (x +y \right ) \cos \left (x +y \right ) \left (x a_{2}+y a_{3}+a_{1}\right )+2 \sin \left (x +y \right ) \cos \left (x +y \right ) \left (x b_{2}+y b_{3}+b_{1}\right ) = 0 \end{equation}

Putting the above in normal form gives

\[ -\sin \left (x +y \right )^{4} a_{3}+2 \sin \left (x +y \right ) \cos \left (x +y \right ) x a_{2}+2 \sin \left (x +y \right ) \cos \left (x +y \right ) x b_{2}+2 \sin \left (x +y \right ) \cos \left (x +y \right ) y a_{3}+2 \sin \left (x +y \right ) \cos \left (x +y \right ) y b_{3}+\sin \left (x +y \right )^{2} a_{2}-\sin \left (x +y \right )^{2} b_{3}+2 \sin \left (x +y \right ) \cos \left (x +y \right ) a_{1}+2 \sin \left (x +y \right ) \cos \left (x +y \right ) b_{1}+b_{2} = 0 \]

Setting the numerator to zero gives

\begin{equation} \tag{6E} -\sin \left (x +y \right )^{4} a_{3}+2 \sin \left (x +y \right ) \cos \left (x +y \right ) x a_{2}+2 \sin \left (x +y \right ) \cos \left (x +y \right ) x b_{2}+2 \sin \left (x +y \right ) \cos \left (x +y \right ) y a_{3}+2 \sin \left (x +y \right ) \cos \left (x +y \right ) y b_{3}+\sin \left (x +y \right )^{2} a_{2}-\sin \left (x +y \right )^{2} b_{3}+2 \sin \left (x +y \right ) \cos \left (x +y \right ) a_{1}+2 \sin \left (x +y \right ) \cos \left (x +y \right ) b_{1}+b_{2} = 0 \end{equation}

Simplifying the above gives

\begin{equation} \tag{6E} b_{2}-\frac {3 a_{3}}{8}+\frac {a_{2}}{2}-\frac {b_{3}}{2}+\frac {a_{3} \cos \left (2 x +2 y \right )}{2}-\frac {a_{3} \cos \left (4 x +4 y \right )}{8}+x a_{2} \sin \left (2 x +2 y \right )+x b_{2} \sin \left (2 x +2 y \right )+y a_{3} \sin \left (2 x +2 y \right )+y b_{3} \sin \left (2 x +2 y \right )-\frac {a_{2} \cos \left (2 x +2 y \right )}{2}+\frac {b_{3} \cos \left (2 x +2 y \right )}{2}+a_{1} \sin \left (2 x +2 y \right )+b_{1} \sin \left (2 x +2 y \right ) = 0 \end{equation}

Looking at the above PDE shows the following are all the terms with \(\{x, y\}\) in them.

\[ \{x, y, \cos \left (2 x +2 y \right ), \cos \left (4 x +4 y \right ), \sin \left (2 x +2 y \right )\} \]

The following substitution is now made to be able to collect on all terms with \(\{x, y\}\) in them

\[ \{x = v_{1}, y = v_{2}, \cos \left (2 x +2 y \right ) = v_{3}, \cos \left (4 x +4 y \right ) = v_{4}, \sin \left (2 x +2 y \right ) = v_{5}\} \]

The above PDE (6E) now becomes

\begin{equation} \tag{7E} b_{2}-\frac {3}{8} a_{3}+\frac {1}{2} a_{2}-\frac {1}{2} b_{3}+\frac {1}{2} a_{3} v_{3}-\frac {1}{8} a_{3} v_{4}+v_{1} a_{2} v_{5}+v_{1} b_{2} v_{5}+v_{2} a_{3} v_{5}+v_{2} b_{3} v_{5}-\frac {1}{2} a_{2} v_{3}+\frac {1}{2} b_{3} v_{3}+a_{1} v_{5}+b_{1} v_{5} = 0 \end{equation}

Collecting the above on the terms \(v_i\) introduced, and these are

\[ \{v_{1}, v_{2}, v_{3}, v_{4}, v_{5}\} \]

Equation (7E) now becomes

\begin{equation} \tag{8E} b_{2}-\frac {3 a_{3}}{8}+\frac {a_{2}}{2}-\frac {b_{3}}{2}+\left (\frac {a_{3}}{2}-\frac {a_{2}}{2}+\frac {b_{3}}{2}\right ) v_{3}-\frac {a_{3} v_{4}}{8}+\left (a_{1}+b_{1}\right ) v_{5}+\left (a_{2}+b_{2}\right ) v_{5} v_{1}+\left (a_{3}+b_{3}\right ) v_{5} v_{2} = 0 \end{equation}

Setting each coefficients in (8E) to zero gives the following equations to solve

\begin{align*} -\frac {a_{3}}{8}&=0\\ a_{1}+b_{1}&=0\\ a_{2}+b_{2}&=0\\ a_{3}+b_{3}&=0\\ \frac {a_{3}}{2}-\frac {a_{2}}{2}+\frac {b_{3}}{2}&=0\\ b_{2}-\frac {3 a_{3}}{8}+\frac {a_{2}}{2}-\frac {b_{3}}{2}&=0 \end{align*}

Solving the above equations for the unknowns gives

\begin{align*} a_{1}&=-b_{1}\\ a_{2}&=0\\ a_{3}&=0\\ b_{1}&=b_{1}\\ b_{2}&=0\\ b_{3}&=0 \end{align*}

Substituting the above solution in the anstaz (1E,2E) (using \(1\) as arbitrary value for any unknown in the RHS) gives

\begin{align*} \xi &= -1 \\ \eta &= 1 \\ \end{align*}

Shifting is now applied to make \(\xi =0\) in order to simplify the rest of the computation

\begin{align*} \eta &= \eta - \omega \left (x,y\right ) \xi \\ &= 1 - \left (-\sin \left (x +y \right )^{2}\right ) \left (-1\right ) \\ &= 1-\sin \left (x \right )^{2} \cos \left (y \right )^{2}-2 \sin \left (x \right ) \cos \left (y \right ) \cos \left (x \right ) \sin \left (y \right )-\cos \left (x \right )^{2} \sin \left (y \right )^{2}\\ \xi &= 0 \end{align*}

The next step is to determine the canonical coordinates \(R,S\). The canonical coordinates map \(\left ( x,y\right ) \to \left ( R,S \right )\) where \(\left ( R,S \right )\) are the canonical coordinates which make the original ode become a quadrature and hence solved by integration.

The characteristic pde which is used to find the canonical coordinates is

\begin{align*} \frac {d x}{\xi } &= \frac {d y}{\eta } = dS \tag {1} \end{align*}

The above comes from the requirements that \(\left ( \xi \frac {\partial }{\partial x} + \eta \frac {\partial }{\partial y}\right ) S(x,y) = 1\). Starting with the first pair of ode’s in (1) gives an ode to solve for the independent variable \(R\) in the canonical coordinates, where \(S(R)\). Since \(\xi =0\) then in this special case

\begin{align*} R = x \end{align*}

\(S\) is found from

\begin{align*} S &= \int { \frac {1}{\eta }} dy\\ &= \int { \frac {1}{1-\sin \left (x \right )^{2} \cos \left (y \right )^{2}-2 \sin \left (x \right ) \cos \left (y \right ) \cos \left (x \right ) \sin \left (y \right )-\cos \left (x \right )^{2} \sin \left (y \right )^{2}}} dy \end{align*}

Which results in

\begin{align*} S&= \tan \left (x +y \right ) \end{align*}

Now that \(R,S\) are found, we need to setup the ode in these coordinates. This is done by evaluating

\begin{align*} \frac {dS}{dR} &= \frac { S_{x} + \omega (x,y) S_{y} }{ R_{x} + \omega (x,y) R_{y} }\tag {2} \end{align*}

Where in the above \(R_{x},R_{y},S_{x},S_{y}\) are all partial derivatives and \(\omega (x,y)\) is the right hand side of the original ode given by

\begin{align*} \omega (x,y) &= -\sin \left (x +y \right )^{2} \end{align*}

Evaluating all the partial derivatives gives

\begin{align*} R_{x} &= 1\\ R_{y} &= 0\\ S_{x} &= \sec \left (x +y \right )^{2}\\ S_{y} &= \sec \left (x +y \right )^{2} \end{align*}

Substituting all the above in (2) and simplifying gives the ode in canonical coordinates.

\begin{align*} \frac {dS}{dR} &= 1\tag {2A} \end{align*}

We now need to express the RHS as function of \(R\) only. This is done by solving for \(x,y\) in terms of \(R,S\) from the result obtained earlier and simplifying. This gives

\begin{align*} \frac {dS}{dR} &= 1 \end{align*}

The above is a quadrature ode. This is the whole point of Lie symmetry method. It converts an ode, no matter how complicated it is, to one that can be solved by integration when the ode is in the canonical coordiates \(R,S\).

Since the ode has the form \(\frac {d}{d R}S \left (R \right )=f(R)\), then we only need to integrate \(f(R)\).

\begin{align*} \int {dS} &= \int {1\, dR}\\ S \left (R \right ) &= R + c_3 \end{align*}

To complete the solution, we just need to transform the above back to \(x,y\) coordinates. This results in

\begin{align*} \tan \left (x +y\right ) = x +c_3 \end{align*}

Which gives

\begin{align*} y = -x +\arctan \left (x +c_3 \right ) \end{align*}

The following diagram shows solution curves of the original ode and how they transform in the canonical coordinates space using the mapping shown.

Original ode in \(x,y\) coordinates	Canonical coordinates transformation	ODE in canonical coordinates \((R,S)\)
\( \frac {dy}{dx} = -\sin \left (x +y \right )^{2}\)		\( \frac {d S}{d R} = 1\)
	\(\!\begin {aligned} R&= x\\ S&= \tan \left (x +y \right ) \end {aligned} \)

Figure 110: Slope field plot
\(y^{\prime }+\sin \left (x +y\right )^{2} = 0\)

1.31.2 Solved as first order ode of type dAlembert

Time used: 1.845 (sec)

Let \(p=y^{\prime }\) the ode becomes

\begin{align*} p +\sin \left (x +y \right )^{2} = 0 \end{align*}

Solving for \(y\) from the above results in

\begin{align*} y &= -x +\arcsin \left (\sqrt {-p}\right )\tag {1A}\\ y &= -x -\arcsin \left (\sqrt {-p}\right )\tag {2A} \end{align*}

This has the form

\begin{align*} y=xf(p)+g(p)\tag {*} \end{align*}

Where \(f,g\) are functions of \(p=y'(x)\). Each of the above ode’s is dAlembert ode which is now solved.

Solving ode 1A

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

\begin{align*} p &= f+(x f'+g') \frac {dp}{dx}\\ p-f &= (x f'+g') \frac {dp}{dx}\tag {2} \end{align*}

Comparing the form \(y=x f + g\) to (1A) shows that

\begin{align*} f &= -1\\ g &= \arcsin \left (\sqrt {-p}\right ) \end{align*}

Hence (2) becomes

\begin{align*} p +1 = -\frac {p^{\prime }\left (x \right )}{2 \sqrt {-p}\, \sqrt {p +1}}\tag {2A} \end{align*}

The singular solution is found by setting \(\frac {dp}{dx}=0\) in the above which gives

\begin{align*} p +1 = 0 \end{align*}

Solving the above for \(p\) results in

\begin{align*} p_{1} &=-1 \end{align*}

Substituting these in (1A) and keeping singular solution that verifies the ode gives

\begin{align*} y = -x +\frac {\pi }{2} \end{align*}

The general solution is found when \( \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}x}}\neq 0\). From eq. (2A). This results in

\begin{align*} p^{\prime }\left (x \right ) = -2 \left (p \left (x \right )+1\right )^{{3}/{2}} \sqrt {-p \left (x \right )}\tag {3} \end{align*}

This ODE is now solved for \(p \left (x \right )\). No inversion is needed. Integrating gives

\begin{align*} \int -\frac {1}{2 \left (p +1\right )^{{3}/{2}} \sqrt {-p}}d p &= dx\\ \frac {\sqrt {-p}}{\sqrt {p +1}}&= x +c_1 \end{align*}

Singular solutions are found by solving

\begin{align*} -2 \left (p +1\right )^{{3}/{2}} \sqrt {-p}&= 0 \end{align*}

for \(p \left (x \right )\). This is because we had to divide by this in the above step. This gives the following singular solution(s), which also have to satisfy the given ODE.

\begin{align*} p \left (x \right ) = -1\\ p \left (x \right ) = 0 \end{align*}

Solving for \(p \left (x \right )\) from the above solution(s) gives (after possible removing of solutions that do not verify)

\begin{align*} p \left (x \right )&=-1\\ p \left (x \right )&=0\\ p \left (x \right )&=-\frac {\left (x +c_1 \right )^{2}}{c_1^{2}+2 c_1 x +x^{2}+1} \end{align*}

Substituing the above solution for \(p\) in (2A) gives

\begin{align*} y = -x +\frac {\pi }{2}\\ y = -x\\ y = -x +\arcsin \left (\sqrt {\frac {\left (x +c_1 \right )^{2}}{c_1^{2}+2 c_1 x +x^{2}+1}}\right )\\ \end{align*}

Solving ode 2A

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

\begin{align*} p &= f+(x f'+g') \frac {dp}{dx}\\ p-f &= (x f'+g') \frac {dp}{dx}\tag {2} \end{align*}

Comparing the form \(y=x f + g\) to (1A) shows that

\begin{align*} f &= -1\\ g &= -\arcsin \left (\sqrt {-p}\right ) \end{align*}

Hence (2) becomes

\begin{align*} p +1 = \frac {p^{\prime }\left (x \right )}{2 \sqrt {-p}\, \sqrt {p +1}}\tag {2A} \end{align*}

The singular solution is found by setting \(\frac {dp}{dx}=0\) in the above which gives

\begin{align*} p +1 = 0 \end{align*}

Solving the above for \(p\) results in

\begin{align*} p_{1} &=-1 \end{align*}

Substituting these in (1A) and keeping singular solution that verifies the ode gives

\begin{align*} y = -x -\frac {\pi }{2} \end{align*}

The general solution is found when \( \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}x}}\neq 0\). From eq. (2A). This results in

\begin{align*} p^{\prime }\left (x \right ) = 2 \left (p \left (x \right )+1\right )^{{3}/{2}} \sqrt {-p \left (x \right )}\tag {3} \end{align*}

This ODE is now solved for \(p \left (x \right )\). No inversion is needed. Integrating gives

\begin{align*} \int \frac {1}{2 \left (p +1\right )^{{3}/{2}} \sqrt {-p}}d p &= dx\\ -\frac {\sqrt {-p}}{\sqrt {p +1}}&= x +c_2 \end{align*}

Singular solutions are found by solving

\begin{align*} 2 \left (p +1\right )^{{3}/{2}} \sqrt {-p}&= 0 \end{align*}

for \(p \left (x \right )\). This is because we had to divide by this in the above step. This gives the following singular solution(s), which also have to satisfy the given ODE.

\begin{align*} p \left (x \right ) = -1\\ p \left (x \right ) = 0 \end{align*}

Solving for \(p \left (x \right )\) from the above solution(s) gives (after possible removing of solutions that do not verify)

\begin{align*} p \left (x \right )&=-1\\ p \left (x \right )&=0\\ p \left (x \right )&=-\frac {\left (x +c_2 \right )^{2}}{c_2^{2}+2 c_2 x +x^{2}+1} \end{align*}

Substituing the above solution for \(p\) in (2A) gives

\begin{align*} y = -x -\frac {\pi }{2}\\ y = -x\\ y = -x -\arcsin \left (\sqrt {\frac {\left (x +c_2 \right )^{2}}{c_2^{2}+2 c_2 x +x^{2}+1}}\right )\\ \end{align*}

The solution

\[ y = -x \]

was found not to satisfy the ode or the IC. Hence it is removed. The solution

\[ y = -x \]

was found not to satisfy the ode or the IC. Hence it is removed.

Figure 111: Slope field plot
\(y^{\prime }+\sin \left (x +y\right )^{2} = 0\)

1.31.3 Maple step by step solution

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

1.31.4 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 C 
1st order, trying the canonical coordinates of the invariance group 
<- 1st order, canonical coordinates successful 
<- homogeneous successful`
 

1.31.5 Maple dsolve solution

Solving time : 0.017 (sec)
Leaf size : 16

dsolve(diff(y(x),x)+sin(x+y(x))^2 = 0, 
       y(x),singsol=all)
 

\[ y = -x -\arctan \left (-x +c_1 \right ) \]

1.31.6 Mathematica DSolve solution

Solving time : 0.165 (sec)
Leaf size : 27

DSolve[{D[y[x],x]+Sin[x+y[x]]^2==0,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}[2 (\tan (y(x)+x)-\arctan (\tan (y(x)+x)))+2 y(x)=c_1,y(x)] \]