DIFFERENTIAL EQUATIONS
CERTIFICATION EVALUATION EXAMS SET
FULL QUESTIONS AND COMPLETE
SOLUTION
●● What is cosh(x)
Answer: (e^x+e^-x)/2
●● What is sinh(x)
Answer: (e^x-e^-x)/2
●● Definition of an ordinary differential equation
Answer: Equations where the derivative is taken with respect to only
ONE veriable
●● Definition of a partial differential equation
Answer: Equations that depend on partial derivatives of several variables
●● Definition of a linear differential equation
Answer: If the dependent variable and its derivatives appear only as first
powers
,●● How is a homogeneous linear equation defined?
Answer: If all terms depend on the dependent variable
●● When is a differential equation linear?
Answer: If no y or y' appears to a power other than 1 and no "term"
contains any of these multiplied together.
●● General n-th order linear DE, largest derivative is n
Answer: a_n(x)d^ny/dx^n+a_n-1(x)d^n-1y/dx^n-1+....
●● What is a partial differential equation and example?
Answer: One that involves partial derivatives, the heat equation is an
example: du/dt=d^2u/dt^2
●● How is an autonomous differential equation defined?
Answer: One that does not involve in independent variable: eg,
dy/dx=f(x,y) (meaning that the change in position only depends on the
function!).
●● General solution to equations of the form d^2y/dx^2=-k^2*y
Answer: C_1*cos(kx)+C_2*sin(kx), where C_1 and C_2 are constants
●● Picard's theorem
, Answer: If a function is near the point (x_0, y_0), f(x,y) is continuous,
and δf/δy exists and is continuous, then there is a solution to y'=f(x,y),
y(x_0) = y_0. This solution exists and is unique.
●● What is the definition of speration of variables?
Answer: A differential equation of the form dy/dx=g(x)/h(x) is
separable. L
●● Test for homogenous functions
Answer: If a function f(x,y) has the property that for all real #'s t, f(tx,
ty)=t^n*f(x,y), for some real number n, then f is a homogenous function
of degree n! (Essentially, substitute in some t for x and y and see if, after
applying the formula's through, you can factor out a t raised to a power
equal to the degree of the function.
●● Definition of a homogenous function
Answer: A differential equation is homogenous if it can be written as:
M(x,y)dx + N(x,y)dy=0 where M and N are homogenous of the same
order.
●● What is the total differential?
Answer: δF/dx dx +δF/dy dy = dF (dF is a function of x and y)
●● Condition for conservative function to be path independant
CERTIFICATION EVALUATION EXAMS SET
FULL QUESTIONS AND COMPLETE
SOLUTION
●● What is cosh(x)
Answer: (e^x+e^-x)/2
●● What is sinh(x)
Answer: (e^x-e^-x)/2
●● Definition of an ordinary differential equation
Answer: Equations where the derivative is taken with respect to only
ONE veriable
●● Definition of a partial differential equation
Answer: Equations that depend on partial derivatives of several variables
●● Definition of a linear differential equation
Answer: If the dependent variable and its derivatives appear only as first
powers
,●● How is a homogeneous linear equation defined?
Answer: If all terms depend on the dependent variable
●● When is a differential equation linear?
Answer: If no y or y' appears to a power other than 1 and no "term"
contains any of these multiplied together.
●● General n-th order linear DE, largest derivative is n
Answer: a_n(x)d^ny/dx^n+a_n-1(x)d^n-1y/dx^n-1+....
●● What is a partial differential equation and example?
Answer: One that involves partial derivatives, the heat equation is an
example: du/dt=d^2u/dt^2
●● How is an autonomous differential equation defined?
Answer: One that does not involve in independent variable: eg,
dy/dx=f(x,y) (meaning that the change in position only depends on the
function!).
●● General solution to equations of the form d^2y/dx^2=-k^2*y
Answer: C_1*cos(kx)+C_2*sin(kx), where C_1 and C_2 are constants
●● Picard's theorem
, Answer: If a function is near the point (x_0, y_0), f(x,y) is continuous,
and δf/δy exists and is continuous, then there is a solution to y'=f(x,y),
y(x_0) = y_0. This solution exists and is unique.
●● What is the definition of speration of variables?
Answer: A differential equation of the form dy/dx=g(x)/h(x) is
separable. L
●● Test for homogenous functions
Answer: If a function f(x,y) has the property that for all real #'s t, f(tx,
ty)=t^n*f(x,y), for some real number n, then f is a homogenous function
of degree n! (Essentially, substitute in some t for x and y and see if, after
applying the formula's through, you can factor out a t raised to a power
equal to the degree of the function.
●● Definition of a homogenous function
Answer: A differential equation is homogenous if it can be written as:
M(x,y)dx + N(x,y)dy=0 where M and N are homogenous of the same
order.
●● What is the total differential?
Answer: δF/dx dx +δF/dy dy = dF (dF is a function of x and y)
●● Condition for conservative function to be path independant
