Solving First Order ODEs using Symmetries 

April 26, 2012 

NAS Research Cluster Spring 2012: 

Hannah Gilmore  (gilmorehf@appstate.edu) 

Michael Kelley     (kelleyma1@appstate.edu) 

Hadi Morrow       (alsoqih@appstate.edu) 

Huy Tu                 (tuhq@appstate.edu) 

Mentor: 

Dr. Bill Cook (cookwj@appstate.edu) 

We looked at solving first order differential equations using symmetry methods. This maple worksheet 

contains a few illustrative examples.  

Initialize: 

>	restart; with(DEtools,DEplot): with(plots):

 

We studied only first order equations which can be solved explicitly for . Any such equation can be  

written in the following form: . We then write as a fraction /. (There are  

many ways to do this. We usually choose and .) 

>	DEqn := diff(y(x),x)=B(x,y(x))/A(x,y(x)): DEqn;

 

(1)

 

In Pfaffian form we have: 

>	-B(x,y)*dx+A(x,y)*dy=0;

 

(2)

 

Example 1: Our first example comes from Peter Hydon's Symmetry Methods for Differential Equations:  

                       A Beginner's Guide. In fact, it is Equation 1.16 (page 15). 

>	B := (x,y) -> (y^3+x^2*y-y-x)/(x^3+x*y^2+y-x): A := (x,y) -> 1: DEqn;

 

(3)

 

This is equation is of the form: which is found in Table 6.1 (page 158) in Brian Cantwell's  

Introduction to Symmetry Analysis. Such equation's solutions have rotational symmetry. The corresponding  

tangent vectors are... 

>	xi := y; eta := -x;

 

 

	(4)

 

So we get the following integrating factor which allows one to solve the DE: 

>	M := 1/(A(x,y)*eta-B(x,y)*xi);

 

(5)

 

>	Int(simplify(-B(x,y)*M),x)=int(-B(x,y)*M,x); Int(simplify(A(x,y)*M),y)=int(A(x,y)*M,y);

 

 

	(6)

 

Notice that this is what Maple is doing internally. 

>	infolevel[dsolve]:= 3: dsolve(DEqn,implicit); infolevel[dsolve]:= 1:

 

 

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 Chini differential order: 1; looking for linear symmetries 1st order, trying the canonical coordinates of the invariance group -> Computing canonical coordinates for the symmetry [y, -x]   -> Calling odsolve with the ODE diff(y(x) x) = -x/y(x) y(x)      *** Sublevel 2 ***      Methods for first order ODEs:      --- Trying classification methods ---      trying a quadrature      trying 1st order linear      trying Bernoulli      <- Bernoulli successful

 

-> Computing canonical coordinates for the symmetry [0, (-y^4+(-2*x^2+1)*y^2-x^4+x^2)/(x^3+x*y^2+y-x)]

 

<- 1st order, canonical coordinates successful
	(7)

 

The following plot shows displays the rotational symmetry of the solutions. 

>

p[1] := dsolve({y(0) = 1, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -1 .. 1): p[2] := dsolve({y(0) = -1, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -1 .. 1): p[3] := dsolve({y(0) = 2.7/5, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -0.95500121 .. 0.19882047): p[4] := dsolve({y(0) = -2.7/5, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -0.19882047 .. 0.95500121): p[5] := dsolve({y(0.19) = 0.1, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -0.0090753013 .. 0.20164819): p[6] := dsolve({y(-0.19) = -0.1, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -0.20164819 .. 0.0090753013): p[7] := dsolve({y(-0.95500121) = -0.085, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -0.95500121 .. 0.19882047): p[8] := dsolve({y(0.95500121) = 0.085, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -0.19882047..0.95500121): p[9] := dsolve({y(0) = 2, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -1.0172241 .. 0.1): p[10] := dsolve({y(0) = -2, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -0.1 .. 1.0172241): p[11] := dsolve({y(0) = 0.15, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -0.6786012 .. 0.048796631): p[12] := dsolve({y(0) = -0.15, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -0.048796631 .. 0.6786012): p[13] := dsolve({y(-0.6786012) = -0.31, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -0.6786012 .. 0.99825818): p[14] := dsolve({y(0.6786012) = 0.31, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -0.99825818 .. 0.6786012): p[15] := dsolve({y(0.048796631) = 0.0475, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -0.0021139841 .. 0.048796631): p[16] := dsolve({y(-0.048796631) = -0.0475, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -0.048796631 .. 0.0021139841): p[17] := dsolve({y(0) = 1.1, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -1 .. 2.25): p[18] := dsolve({y(0) = -1.1, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -2 .. 1): p[19] := dsolve({y(1.2) = 0.2, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -1 .. 2.25): p[20] := dsolve({y(-1.2) = -0.2, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -2.25 .. 1): display(seq(odeplot(p[i]),i=1..20),scaling=constrained);

 

Example 2:  

>	B := (x,y) -> y/(x+(y/x)): A := (x,y) -> 1: DEqn := diff(y(x),x)=B(x,y(x))/A(x,y(x)): DEqn;

 

(8)

 

Of the form:  

The table gives us the following: 

>	xi := x*y; eta := y^2;

 

 

	(9)

 

>	M := 1/(A(x,y)*eta-B(x,y)*xi);

 

(10)

 

>	Int(simplify(-B(x,y)*M),x)=int(-B(x,y)*M,x); Int(simplify(A(x,y)*M),y)=int(A(x,y)*M,y);

 

 

	(11)

 

Solution: 

>	C = -1/2*x^2/y^2-1/y;

 

(12)

 

Maple's solution: 

>	infolevel[dsolve]:= 3: dsolve(DEqn); infolevel[dsolve]:= 1:

 

 

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 G

 

<- homogeneous successful
	(13)

 

Example 3:  

>	B := (x,y) -> y/(x+(y/x)^2+1): A := (x,y) -> 1: DEqn := diff(y(x),x)=B(x,y(x))/A(x,y(x)): DEqn;

 

(14)

 

Of the form:  

The table gives us the following: 

>	xi := x*y; eta := y^2;

 

 

	(15)

 

>	M := 1/(A(x,y)*eta-B(x,y)*xi);

 

(16)

 

>	Int(simplify(-B(x,y)*M),x)=int(-B(x,y)*M,x); Int(simplify(A(x,y)*M),y)=int(A(x,y)*M,y);

 

 

	(17)

 

Solution: 

>	C = -x/y+arctan(x/y)-1/y;

 

(18)

 

Maple's solution: 

>	infolevel[dsolve]:= 3: dsolve(DEqn,implicit); infolevel[dsolve]:= 1:

 

 

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 Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE trying 1st order ODE linearizable_by_differentiation

 

 

 

--- Trying Lie symmetry methods, 1st order --- -> Computing symmetries using: way = 2
<- successful computation of symmetries. trying an integrating factor from the invariance group <- integrating factor successful
	(19)

 

A couple of plots of some solutions: 

>

p[1] := dsolve({y(-2) = 0.1, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -2 .. 2): p[2] := dsolve({y(-1.5) = 0.5, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -2 .. 2): p[3] := dsolve({y(-1.05) = 0.5, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -1.08229 .. 2): p[4] := dsolve({y(-1.05) = 0.1, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -1.100457 .. 2): p[5] := dsolve({y(-1.01) = 0.01, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -1.392886 .. 2): p[6] := dsolve({y(-1.4) = 1, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -1.4056867 .. 2): p[7] := dsolve({y(.5) = 3/5, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -1.014050 .. 2): p[8] := dsolve({y(-2) = -0.1, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -2 .. 2): p[9] := dsolve({y(-1.5) = -0.5, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -2 .. 2): p[10] := dsolve({y(-1.05) = -0.5, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -1.08229 .. 2): p[11] := dsolve({y(-1.05) = -0.1, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -1.100457 .. 2): p[12] := dsolve({y(-1.01) = -0.01, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -1.392886 .. 2): p[13] := dsolve({y(-1.4) = -1, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -1.4056867 .. 2): p[14] := dsolve({y(.5) = -3/5, (D(y))(x) = rhs(DEqn)}, type = numeric, range = -1.014050 .. 2): display(seq(odeplot(p[i]),i=1..14),scaling=constrained);

 

>	S := dsolve(DEqn,implicit);

 

(20)

 

>	contourplot(lhs(subs(_C1=0,S)),x=-1.8..2,y=-2.5..2.5,contours=[-0.1,-0.5,-0.7,-1,-2,-5,0.1,0.5,0.7,1,2,5],numpoints=5000,color=red);

 

Example 4:  

>	B := (x,y) -> (y/x)+x*((y/x)^2+1): A := (x,y) -> 1: DEqn := diff(y(x),x)=B(x,y(x))/A(x,y(x)): DEqn;

 

(21)

 

This is of the form  

The table then gives us... 

>	xi := 1; eta := y/x;

 

 

	(22)

 

>	M := 1/(A(x,y)*eta-B(x,y)*xi);

 

(23)

 

>	Int(simplify(-B(x,y)*M),x)=int(-B(x,y)*M,x); Int(simplify(A(x,y)*M),y)=int(A(x,y)*M,y);

 

 

	(24)

 

Solution: 

>	C=x+arctan(x/y);

 

(25)

 

Maple's solution: 

>	infolevel[dsolve]:= 3: dsolve(DEqn,implicit); infolevel[dsolve]:= 1:

 

 

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
	(26)

 

Example 5:  

>	B := (x,y) -> y/(x*(ln(x)+ln(y))): A := (x,y) -> 1: DEqn := diff(y(x),x)=B(x,y(x))/A(x,y(x)): DEqn;

 

(27)

 

This equation is of the form  

The table then gives us... 

>	xi := x*y; eta := 0;

 

 

	(28)

 

>	M := 1/(A(x,y)*eta-B(x,y)*xi);

 

(29)

 

>	Int(simplify(-B(x,y)*M),x)=int(-B(x,y)*M,x); Int(simplify(A(x,y)*M),y)=int(A(x,y)*M,y);

 

 

	(30)

 

Solution: 

>	C=ln(x)/y+ln(y)/y+1/y;

 

(31)

 

Maple's solution: 

>	infolevel[dsolve]:= 3: dsolve(DEqn,implicit); infolevel[dsolve]:= 1:

 

 

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 G 1st order, trying the canonical coordinates of the invariance group -> Computing canonical coordinates for the symmetry [0, -(ln(x)+ln(y)+1)/(ln(x)+ln(y))*y] <- 1st order, canonical coordinates successful <- homogeneous successful

(32)

 

Example 6: 

>	B := (x,y) -> y*(ln(y)-x): A := (x,y) -> 1: DEqn := diff(y(x),x)=B(x,y(x))/A(x,y(x)): DEqn;

 

(33)

 

This equation is of the form  

The table then gives us... 

>	xi := 1; eta := y;

 

 

	(34)

 

>	M := 1/(A(x,y)*eta-B(x,y)*xi);

 

(35)

 

>	Int(simplify(-B(x,y)*M),x)=int(-B(x,y)*M,x); Int(simplify(A(x,y)*M),y)=int(A(x,y)*M,y);

 

 

	(36)

 

Solution: 

>	C=x-ln(ln(y)-x-1);

 

(37)

 

Maple's solution: 

>	infolevel[dsolve]:= 3: dsolve(DEqn,implicit); infolevel[dsolve]:= 1:

 

 

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 Chini differential order: 1; looking for linear symmetries

 

1st order, trying the canonical coordinates of the invariance group -> Computing canonical coordinates for the symmetry [1, y]   -> Calling odsolve with the ODE diff(y(x) x) = y(x) y(x)      *** Sublevel 2 ***      Methods for first order ODEs:      --- Trying classification methods ---      trying a quadrature      trying 1st order linear      <- 1st order linear successful -> Computing canonical coordinates for the symmetry [0, (x+1-ln(y))*y] <- 1st order, canonical coordinates successful

(38)