Its invariant lines are y+ 2 with co-factor 1; 2x+ 2y+ 3 with co-factor 2. Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Its invariant lines are y+ 2 with co-factor 1; 2x+ 2y+ 3 with co-factor 2. Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Find `linear andc a scalar, such that
Its invariant lines are y+ 2 with co-factor 1;
2x+ 2y+ 3 with co-factor 2. Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Find `linear andc a scalar, such that F · ∇`=c`.
Its invariant lines are y+ 2 with co-factor 1; 2x+ 2y+ 3 with co-factor 2. Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Find `linear andc a scalar, such that (2x+y+ 1)∂`
∂x + (y+ 2)∂`
∂y =c`.
Its invariant lines are y+ 2 with co-factor 1; 2x+ 2y+ 3 with co-factor 2. Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Find `linear andc a scalar, such that
Its invariant lines are y+ 2 with co-factor 1; 2x+ 2y+ 3 with co-factor 2. Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Find `linear andc a scalar, such that
Its invariant lines are y+ 2 with co-factor 1; 2x+ 2y+ 3 with co-factor 2. Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Find `linear andc a scalar, such that
Solve an eigenvalue problem for the linear operator whose matrix in the basis {x,y,1} is
Its invariant lines are y+ 2 with co-factor 1;
2x+ 2y+ 3 with co-factor 2. Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Find `linear andc a scalar, such that
Solve an eigenvalue problem for the linear operator whose matrix in the basis {x,y,1} is
Its invariant lines are y+ 2 with co-factor 1; 2x+ 2y+ 3 with co-factor 2. Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Solve an eigenvalue problem for the linear operator whose matrix in the basis {x,y,1} is
Its invariant lines are y+ 2 with co-factor 1; 2x+ 2y+ 3 with co-factor 2. Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Solve an eigenvalue problem for the linear operator whose matrix in the basis {x,y,1} is
the matrix the differential equation
eigenvector eigenvalue ` c
Its invariant lines are y+ 2 with co-factor 1; 2x+ 2y+ 3 with co-factor 2. Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Solve an eigenvalue problem for the linear operator whose matrix in the basis {x,y,1} is
the matrix the differential equation
eigenvector eigenvalue ` c
(0,1,2)
Its invariant lines are y+ 2 with co-factor 1; 2x+ 2y+ 3 with co-factor 2. Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Solve an eigenvalue problem for the linear operator whose matrix in the basis {x,y,1} is
the matrix the differential equation
eigenvector eigenvalue ` c
(0,1,2) 1
Its invariant lines are y+ 2 with co-factor 1; 2x+ 2y+ 3 with co-factor 2. Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Solve an eigenvalue problem for the linear operator whose matrix in the basis {x,y,1} is
the matrix the differential equation
eigenvector eigenvalue ` c
(0,1,2) 1 y+ 2
Its invariant lines are y+ 2 with co-factor 1; 2x+ 2y+ 3 with co-factor 2. Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Solve an eigenvalue problem for the linear operator whose matrix in the basis {x,y,1} is
the matrix the differential equation
eigenvector eigenvalue ` c
(0,1,2) 1 y+ 2 1
Its invariant lines are y+ 2 with co-factor 1; 2x+ 2y+ 3 with co-factor 2. Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Solve an eigenvalue problem for the linear operator whose matrix in the basis {x,y,1} is
the matrix the differential equation
eigenvector eigenvalue ` c
(0,1,2) 1 y+ 2 1
(2,2,3)
Its invariant lines are y+ 2 with co-factor 1; 2x+ 2y+ 3 with co-factor 2. Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Solve an eigenvalue problem for the linear operator whose matrix in the basis {x,y,1} is
the matrix the differential equation
eigenvector eigenvalue ` c
(0,1,2) 1 y+ 2 1
(2,2,3) 2
Its invariant lines are y+ 2 with co-factor 1; 2x+ 2y+ 3 with co-factor 2. Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Solve an eigenvalue problem for the linear operator whose matrix in the basis {x,y,1} is
the matrix the differential equation
eigenvector eigenvalue ` c
(0,1,2) 1 y+ 2 1
(2,2,3) 2 2x+ 2y+ 3
Its invariant lines are y+ 2 with co-factor 1; 2x+ 2y+ 3 with co-factor 2. Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Solve an eigenvalue problem for the linear operator whose matrix in the basis {x,y,1} is
the matrix the differential equation
eigenvector eigenvalue ` c
(0,1,2) 1 y+ 2 1
(2,2,3) 2 2x+ 2y+ 3 2
Its invariant lines are y+ 2 with co-factor 1; 2x+ 2y+ 3 with co-factor 2. Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Its invariant lines are y+ 2 with co-factor 1;
2x+ 2y+ 3 with co-factor 2.
Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Its invariant lines are y+ 2 with co-factor 1;
2x+ 2y+ 3 with co-factor 2.
Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations
An example: the Jacobi equation
Suppose that F = (2x+y+ 1,y+ 2).
Its invariant lines are y+ 2 with co-factor 1;
2x+ 2y+ 3 with co-factor 2.
Since 2·1 + (−1)·2 = 0,
h= (y+ 2)2(2x+ 2y+ 3)−1, is a first integral of ˙X =F(X).
S. C. Coutinho Proving the nonexistence of algebraic solutions of differential equations