Differential Equations True and False

Non Homogeneous ODES

-Guess and Check -variation of parameters -greens function

Higher Order Homogeneous ODE

-constant coefficient -Cauchy Euler

Ways to solve 1st order ODE

-separable -linear u=e^int(P(x)) -Bernoulli u=y^1-n

The set of functions {sin² x,cos² x,1} is linearly independent.

False because 1sin^2x+1cos^2x-1=0

Every autonomous ODE has at least one [asymptotically] stable solution.

False let y'=f(y) be the autonomous ODE. Let y* be the equilibrium point. If f'(y*)>0, the equilibrium is unstable

The ODE (xy)dx + (2x² + 3y² −20)dy = 0 is exact.

False because My=x and Nx=4x, since they are not the same they are not exact

The equation y =C/e^x+x²/e^x+ 1 has no transient terms.

False because the limit of C/e^x is a transient term

The set of functions {1,x²,4−3x²} is linearly independent.

False because wronskian comes out to be zero

The ODE dy/dx+ xsiny = 0 is linear.

False it is not linear because of the sin(y)

The set of functions {1,x,ex} is linearly dependent.

False, W is not equal to 0 thus it is linearly independent

If the roots of the auxiliary equation of third order linear ODE with constant coefficients are r = 2, r = 1 + 3i and r = 1−3i , the complementary solution is yc = C1e2x + C2 sin3x + C3 cos3x.

False, because there should be an e in front of sin3x and cos3x

For any two piecewise continuous functions f and g of exponential order, L{f(t)g(t)} = L{f(t)}L{g(t)}

False, because this breaks Laplace rules. You only separate them when two functions are being added together.

If the roots of the the auxiliary equation of a 2nd order Cauchy-Euler ODE are m = 2±i then the fundamental set of the ODE is {x² sinx,x² cosx}.

False, because y=Ax^alpha*cos(beta*ln(x))+Bx^alpha*sin(beta*ln(x))

If two non homogeneous ODEs have the same fundamental set, then they also have the same particular solution

False, because yp is based on the function given

The ODE dy/dt= y(2−y) has a stable solution at y = 0.

False, because zero is semi-stable

The ODE (4x2 −4y2)dx + (4xy−4x2)dy is homogeneous of degree 4.

False, it is homogeneous of degree 2 not 4

The ODE dy/dx −xtany = ex is linear

False, it is not linear because of tan(y)

If f,g are functions with Wronskian W(x) and W(c) = 0 for some c ∈ R, then f,g are linearly dependent.

False, let y1=1 and y2=x² then solve for W(x) which will equal 2x. So it is linearly independent

The Laplace transform is a linear transformation.

False, not all Laplace transforms are linear because you can solve them using the integral definition which isn't linear

If f(t) is differentiable and L{f(t)} exists, then L{f'(t)} = d ds [L{f(t)}].

False, prove with L{1} =1/s; -1/s²≠0

An ODE of the form y''' + P(x)y'' + Q(x)y' + R(x) = f(x) may have up to two initial conditions.

False, you would need three initial conditions because it is a third order ODE

If yp is the particular solution of the ODE a2(x)y'' + a1(x)y' + a0(x)y = g(x); y(0) = 1, y'(0) = 2 then yp(0) = 1.

Guess try your best

The function y = 0 is a solution to every homogeneous ODE.

TRUE

The set of function{ex,ex2,ex3} is the fundamental set of some 3rd order constant coefficient ODE.

TRUE

If y1 and y2 are solutions to second order linear homogeneous ODE, then y = 6y1 −8y2 is also a solution.

True

The ODE x2y2 dx + (x2 + y2)dy = 0 is homogeneous of degree 2.

True

The ODE xtany dx + (½x²sec² y + sinhy)dy = 0 is exact.

True

The ODE x²y''−cosxy' + exy = 0 has a power series solution centered at x = 0.

True

The ODE y'' + (1/ x−1) y' + exy = 0 has a singular point at x = 1.

True

The set of function{x,x³,2x³−x}is the fundamental set of some 3rd order ODE.

True

The set {1,x,x2,x2 −1} is the fundamental set of some fourth order ODE.

True

{e2x,xe2x,x2e2x} is the fundamental set of some 3rd order ODE.

True