Mind-Warm-Ups Ordinary Differential equations

Problem 1: What is a differential equation?
Ans: A differential equation is an equation that contains at least one independent variable, at least one dependent variable, and derivatives of the dependent variables with respect to some or all of the independent variables.

Problem 2: Which of the following is an ordinary differential equation?
(a) d2y/dt2 + ω2y = 0
(b) (1/c2)∂2u/∂t2 = ∂2u/∂x2
Ans: An ordinary differential equation has only one independent variable. Hence the derivatives are all ordinary derivatives.
(a) is an ordinary differential equation because it has only t has its independent variable.

Problem 3: What is the order of the following differential equation?
d2y/dx2 + xdy/dx + (x2 - n2)y = 0
Ans: Order is 2 (order of the highest derivative in the equation).

Problem 4 : Indicate the dependent and independent variables in the following general form of the nth order ordinary differential equation:
F(t, y, dy/dt, ..., dny/dtn) = 0
Ans: t is the independent variable while y is the dependent variable.

Problem 5 : What condition is necessary and sufficient for F(t, y, dy/dt, ..., dn/dtn) = 0 to be a linear ordinary differential equation?
Ans: F(t, y, dy/dt, ..., dny/dt) = a0(t)dny/dtn + a1(t)dn-1y/dtn-1 + ... + an-1(t)dy/dt + an(t)y + f(t).

Problem 6: Is the following differential equation linear?
ydy/dt + y = f(t)
Ans: No. Figure out why.

Problem 7: What does the following statements mean with respect to differential equations?
(a) A solution exists
(b) The solution is unique
(c) The problem is overdetermined
(d) The problem is underdetermined
Ans:(a) There is at least one function which satisfies all the conditions upon which the differential equation is based
(b) There is only one function which satisfies the conditions
(c) Too many conditions to be met and therefore no solution
(d) There are not enough conditions for a unique solution

Problem 8: Indicate the order and linearity of the following ODEs:
(a) dy/dtd2y/dt2 = y
(b) (1-x2)d2y/dx2 - 2xdy/dx + n(n+1)y = 0; where n is a constant.
(c) eyd2y/dx2 + x3 = 0
(d) sin(y) + xcos(dy/dx) = 0
Ans: Only (b) is linear ODE. The rest are nonlinear.

Problem 9: What is an initial-value problem with respect to ordinary differential equations?
Ans:An initial-value problem is the case where all the specified values of the solution of the ODE and the derivatives of the ODE are given at a point.

Problem 10: What is a boundary-value problem with respect to ordinary differential equations?
Ans: A boundary-value problem is the case where data are given for the solution at more than one point.

Peter Oye Sagay

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