MATH 252 F Test 1

Theorem 3.5: Orthogonal Trajectory Theorem

Let c∈R. Suppose that F,G:D→R,where D⊂ R3. Assume that F(x,y,c) = 0 and G(x,y,c) = 0 are equations of paths in R^2 with well-defined tangent lines at each point on these paths. Then G subscript C (Gc) is an orthogonal family of trajectories for the family F subscript C (Fc) iff for each Gc∈Gc, we have that G(x,y,c) = 0 is a solution to the ode given by dy/dx=−1/f(x,y),where (x,y)∈Fc∩Gc and f(x,y) is the slope of the tangent line to Fc at (x,y).

Theorem 10.6 : One-Variable Transformation Theorem for Homogeneous Functions of Two Variables

Suppose that D⊂R^2, and assume that ∀(x,y)∈D,x does not = 0. Let f:D→R. Then f is homogeneous iff ∃E⊂R and F:E→R such that f(x,y) =F(y/x)∀(x,y)∈D.

Theorem 14.1 :Second-0rder to First-Order System Reduction Theorem

Suppose that J is a non-degenerate interval in R, and assume that y : J → R is twice-differentiable and unknown. Let D: = { (x,y(x) , y′(x)) : x ∈ J }. Let F : D → R. Then the system dy/dx = v, dv/dx = F(x,y,v)i s a solution to y'′=F(x,y,y′).

Theorem 12.13 : Integrating Factor Characterization Theorem for ODE's of the form M(x,y)dx + N(x,y) dy= 0.

Suppose that J is an open interval in R and that y : J → R is differentiable and unknown. Let { (x,y(x)) : x∈J }⊂ D ⊂ R^2 . Suppose that M,N:D → R. Then I is an integrating factor of M(x,y)dx + N(x,y)dy = 0 iff I is a solution to N(x,y)(I subscript x)(x,y)−M(x,y)Iy (I subscript y)(x,y) = [(M subscript y)(x,y)− (N subscript x)(x,y)] I(x,y).

Theorem 12.15: One Variable INtegration Factor Generation Theorem for ODE's of the form M(x,y)dx + N(x,y)dy = 0.

Suppose that J is an open interval in R and that y : J → R is differentiable and unknown. Let {(x,y(x)) :x ∈J }⊂ D ⊂ R^2 . Suppose that M,N: D → R. Then the following two statements hold: (1) ∃f : J → R such that ∀(x,y) ∈ D, we have that f(x) =M subscript y(x,y)−N subscript x(x,y)N(x,y), iff I = e^F is an integrating factor of M(x,y)dx + N(x,y)dy = 0 for some F∈∫f(x)dx. (2) ∃g : Rng(y) → R such that ∀(x,y) ∈ D, we have that g(y) =M subscript y(x,y)− N subscript x(x,y)M(x,y),iff I:=e^−G is an integrating factor of M(x,y)dx + N(x,y)dy = 0 for some G∈∫g(x)dx.

Theorem 12.11: Integrating Factor Solution Invariance Theorem

Suppose that J is an open interval in R and that y: J → R is differentiable and unknown. Let { (x,y(x)) : x ∈J }⊂D⊂R^2. Suppose that M,N:D → R. Suppose that I is an integrating factor for M(x,y)dx + N(x,y)dy = 0, and that an equation or system of equations is a solution to I(x,y)M(x,y)dx + I(x,y)N(x,y)dy = 0. Then this solution is also a solution of M(x,y)dx + N(x,y)dy= 0.

Theorem 12.7: Exactness Test

Suppose that J is an open interval in R and that y: J → R is differentiable and unknown. Let { (x,y(x)) : x∈J ⊂ D ⊂ R^2. Suppose that M,N:D → R and that D is simply connected. Then M(x,y)dx + N(x,y)dy = 0 is exact iff M subscript y(My)=N subscript x (Nx)

Theorem 12.5: Exact ODE Potential Solution Theorem

Suppose that J is an open interval in R and that y: J→ R is differentiable and unknown. Let D ={(x,y(x)) :x∈J}.Suppose that M,N:D → R and that M(x,y)dx + N(x,y)dy = 0 is exact. Then an equation is a solution of this ode iff the equation has the property that ∃c∈R and a potential φ for this ode, such that the equation can be rewritten as φ(x,y) =c.

Theorem 6.1: Solution Existence and Uniqueness Theorem for nth Order Linear IVP's w/ Continuous Coefficients

Suppose that n∑k=0an−ky(n−k)=f, is an nth order and linear ODE in the unknown function y on a non-degenerate interval "I" upon which a subscript j,∀j∈ {0,1,2,...,n}, and f are real-valued and continuous functions. Then there is a unique solution y of the IVP consisting of the above ode together with the initial conditions y superscript (k) times (x0) =y subscript k for each k∈{0,1,2,...,n−1}.

Theorem 11.4: Bernoulli ODE Change of Variables Theorem

Suppose that we are given a Bernoulli ode of the form y′+p(x)y=q(x)y^r, where p,q: Dom(y)→R and r∈R. Then (a) The ode is a monic, first order linear ODE if either r= 0 or r= 1. (b) If r doesn't = 0,1, and we let u=y^(1−r), then the ode u′+ (1−r)p(x)u= (1−r)q(x) is a monic, first-order, linear ode in u. (c) A solution to the above ode when r doesn't= 0,1 is in turn given by y=u^(1/(1−r)).

Theorem 7.4: Implicit Solution Theorem for Separable ODE's

Suppose that y is an unknown differentiable, real-valued function of a real variable, and consider p(y)y'=q(x), where p: Rng(y)→R and q: Dom(y)→R. Then ∫p(y)dy=∫q(x)dx is a general solution.

Theorem 10.8: Change of Variables Theorem for First-Order Homogeneous ODE's

Suppose that y: D→ R is an unknown function, and let y′=f(x,y) be homogeneous. Assume that 0 does not ∈D, and let E⊂R and F:E→R be such tha tf(x,y) =F(y/x)∀(x,y)∈ D. Let V=y/x. Then both of the following statements hold: (a)1/F(V)−V *V′=1/x is a separable ode.(b) A solution to y′=f(x,y) is y=V*x, where V is a solution of the above ode.

Theorem 9.6: Integrating Factor Theorem for Monic, First-Order, Linear DE'S

Suppose that y: I→R is an unknown function, where I is an open interval in R. Let p,q: I→R. Assume that p is an anti differentiable on I. Assume that if P is an antiderivative of p and v: I→R is given by v(x) =e^P(x), then vq is antidifferentiable. Then f is a solution of y′+p(x)y=q(x) iff there is a C∈R such that f is given by f(x) =1/v(x) *[G(x) +C], where P∈∫p(x)dx, and G∈∫v(x)q(x)dx.