DIFFERENTIAL EQUATIONS 2024 TEST
BANK QUESTIONS WITH ANSWERS
GRADED A+
Linearisation Theorem - Answer-Suppose y_0 is an equilibrium point of an autonomous
differential equation where f(y) is continuously differentiable, then if f'(y_0)<0 then y_0 is
a sink and if f'(y_0)>0 then y_0 is a source.
Phase Plane - Answer-The plane of dependent variables of a system of differential
equations
Damper Harmonic Oscillator - Answer-m(d^2y/dt^2)+b(dy/dt)+ky=0
Linear System of Differential Equations - Answer-(dx/dt)=ax+by, (dy/dt)=cx+dy
Trivial Solution - Answer-Y=(0, 0)
Linearity Principle of Systems, Part 1 - Answer-Suppose (dY/dt)=AY is a linear system
of differential equations. If Y(t) is a solution of the system and k is any constant, then
kY(t) is also a solution.
Linearity Principle of Systems, Part 2 - Answer-Suppose (dY/dt)=AY is a linear system
of differential equations. If Y_1(t) and Y_2(t) are both solutions of the system, then
Y_1(t)+Y_2(t) is a solution as well.
Linearly Independent Solutions - Answer-Suppose Y_1(t) and Y_2(t) are solutions to the
linear system (dY/dt)=AY. If Y_1(0) and Y_2(0) are linearly independent, then for any
initial condition Y(0)=(x_0, y_0) we can find constants k_1 and k_2 such that k_1 Y_1(t)
+ k_2 Y_2(t) is the solution to the IVP.
Straight Line Solution Equation - Answer-Y(t)=e^(λt) V
General Solution to a Linear System of Differential Equations - Answer-Y(t) = k_1
e^(λ_1 t) V_1 + k_2 e^(λ_2 t) V_2
Euler's Formula for Imaginary Exponentials - Answer-e^(a+bi) = e^a (cosb + isinb)
Complex-Valued Solutions Theorem - Answer-Suppose Y(t) is a complex-valued
solution for the linear system (dY/dt)=AY where A is composed of real numbers. If Y(t) =
Y_re(t) + iY_im(t) where Y_re(t) and Y_im(t) are real-valued, then both are solutions of
the system.
BANK QUESTIONS WITH ANSWERS
GRADED A+
Linearisation Theorem - Answer-Suppose y_0 is an equilibrium point of an autonomous
differential equation where f(y) is continuously differentiable, then if f'(y_0)<0 then y_0 is
a sink and if f'(y_0)>0 then y_0 is a source.
Phase Plane - Answer-The plane of dependent variables of a system of differential
equations
Damper Harmonic Oscillator - Answer-m(d^2y/dt^2)+b(dy/dt)+ky=0
Linear System of Differential Equations - Answer-(dx/dt)=ax+by, (dy/dt)=cx+dy
Trivial Solution - Answer-Y=(0, 0)
Linearity Principle of Systems, Part 1 - Answer-Suppose (dY/dt)=AY is a linear system
of differential equations. If Y(t) is a solution of the system and k is any constant, then
kY(t) is also a solution.
Linearity Principle of Systems, Part 2 - Answer-Suppose (dY/dt)=AY is a linear system
of differential equations. If Y_1(t) and Y_2(t) are both solutions of the system, then
Y_1(t)+Y_2(t) is a solution as well.
Linearly Independent Solutions - Answer-Suppose Y_1(t) and Y_2(t) are solutions to the
linear system (dY/dt)=AY. If Y_1(0) and Y_2(0) are linearly independent, then for any
initial condition Y(0)=(x_0, y_0) we can find constants k_1 and k_2 such that k_1 Y_1(t)
+ k_2 Y_2(t) is the solution to the IVP.
Straight Line Solution Equation - Answer-Y(t)=e^(λt) V
General Solution to a Linear System of Differential Equations - Answer-Y(t) = k_1
e^(λ_1 t) V_1 + k_2 e^(λ_2 t) V_2
Euler's Formula for Imaginary Exponentials - Answer-e^(a+bi) = e^a (cosb + isinb)
Complex-Valued Solutions Theorem - Answer-Suppose Y(t) is a complex-valued
solution for the linear system (dY/dt)=AY where A is composed of real numbers. If Y(t) =
Y_re(t) + iY_im(t) where Y_re(t) and Y_im(t) are real-valued, then both are solutions of
the system.
