Bungee Jumping
1. Find two (2) methods (other than Undetermined Coefficients) that can be used to solve higher order ODE. For each method, solve one higher order ODE problem.
Answer:
The general form of an ODE with order nis
Fx, yx, y ' 'x,…,ynx=f(x)
As in the case of second-order ODEs, such an ODE can be classified as linear or nonlinear. The general form of a linear ODE of order is an(x)dnydxn+an-1(x)dn-1ydxn-1+…+a1(x)dydx+a0xy=f(x) If f(x)is the zero function, the equation is said to behomogeneous. Many methods for solving higher order ODEs can be generalized to linear ODEs of ordern, where nis greater than 2. If the order of the ODE is not important, it is simply called a linear ODE.
a). Variation of Parameters
Let’s,
an(x)dnydxn+an-1(x)dn-1ydxn-1+…+a1(x)dydx+a0xy=f(x)
Put it, in the standard form, yn+Pxyn-1+…+Qxy '+Rxy=f(x)
Then, we assume that, yP=u1y1+u2y2 By Cramer’s rule, the solution of the system, y1u1 '+y2u2 '=0 y1 'u1 '+y2 'u2 '=f(x) can be expressed in terms of determinants: u1 '=W1W= -y2 f(x)Wandu2 '=W2W= -y1f(x)W
Where W=y1y2y1y2, W1=0y2f(x)y2.W2=y10y2f(x)
Example;
Solve y ' '-y '-2y=2e-x.
Solution
Form an auxiliary equation. m2-m-2=0 m-2m+1=0 m=2, m=-1
Therefore,
yc=c1e2x+c2e-x
So that, y1=e2x, y2=e-x,f(x)=2e-x,
Next, compute the W.
We2x,e-x=e2xe-x2e2x-e-x= -3ex
W1=0e-x2e-x-e-x=-2e-2x
W2=e2x02e2x2e-x=2ex
So that, u1 '=W1W= -2e-2x-3ex=23e-3x u2 '=W2W= 2ex-3ex=-23
Therefore,
u1=-29e-2x+C1 u2=-23x+C2 So that, yp=-29e-2x-23x So that the full solution is y=c1e2x+c2e-x-29e-2x-23x b) Cauchy-Euler Equation an(x)dnydxn+an-1(x)dn-1ydxn-1+…+a1(x)dydx+a0xy=f(x) where the coefficients an, an-1, . . ., a0 are constants, is known diversely as a Cauchy–Eulerequation, an Euler–Cauchy equation, an Euler equation, or an equidimensional equation. The observable characteristic of this type of equation is that the degree k =n, n –1, … 1, 0 of the monomial coefficients xkmatches the order k of differentiationdkxdyk:
The
Answer:
The general form of an ODE with order nis
Fx, yx, y ' 'x,…,ynx=f(x)
As in the case of second-order ODEs, such an ODE can be classified as linear or nonlinear. The general form of a linear ODE of order is an(x)dnydxn+an-1(x)dn-1ydxn-1+…+a1(x)dydx+a0xy=f(x) If f(x)is the zero function, the equation is said to behomogeneous. Many methods for solving higher order ODEs can be generalized to linear ODEs of ordern, where nis greater than 2. If the order of the ODE is not important, it is simply called a linear ODE.
a). Variation of Parameters
Let’s,
an(x)dnydxn+an-1(x)dn-1ydxn-1+…+a1(x)dydx+a0xy=f(x)
Put it, in the standard form, yn+Pxyn-1+…+Qxy '+Rxy=f(x)
Then, we assume that, yP=u1y1+u2y2 By Cramer’s rule, the solution of the system, y1u1 '+y2u2 '=0 y1 'u1 '+y2 'u2 '=f(x) can be expressed in terms of determinants: u1 '=W1W= -y2 f(x)Wandu2 '=W2W= -y1f(x)W
Where W=y1y2y1y2, W1=0y2f(x)y2.W2=y10y2f(x)
Example;
Solve y ' '-y '-2y=2e-x.
Solution
Form an auxiliary equation. m2-m-2=0 m-2m+1=0 m=2, m=-1
Therefore,
yc=c1e2x+c2e-x
So that, y1=e2x, y2=e-x,f(x)=2e-x,
Next, compute the W.
We2x,e-x=e2xe-x2e2x-e-x= -3ex
W1=0e-x2e-x-e-x=-2e-2x
W2=e2x02e2x2e-x=2ex
So that, u1 '=W1W= -2e-2x-3ex=23e-3x u2 '=W2W= 2ex-3ex=-23
Therefore,
u1=-29e-2x+C1 u2=-23x+C2 So that, yp=-29e-2x-23x So that the full solution is y=c1e2x+c2e-x-29e-2x-23x b) Cauchy-Euler Equation an(x)dnydxn+an-1(x)dn-1ydxn-1+…+a1(x)dydx+a0xy=f(x) where the coefficients an, an-1, . . ., a0 are constants, is known diversely as a Cauchy–Eulerequation, an Euler–Cauchy equation, an Euler equation, or an equidimensional equation. The observable characteristic of this type of equation is that the degree k =n, n –1, … 1, 0 of the monomial coefficients xkmatches the order k of differentiationdkxdyk:
The
References: Zill, Dennis G. 2009. A First Course In Differential Equations with Modelling Applications. 9th Ed. Pacific Grove: Brooks/Cole – Thomson Learning Inc. Blanchard, P., Devaney, R.L., & Hall, G.R. (2002). Diffrential equations. 2nd Ed. Pacific Grove: Brooks/Cole – Thomson Learning Inc. Hollis, S.L (2002). Differntial Equations with Boundary Value Problems. Pearson EducationPrentice-Hall Inc Blanchard, P., Devaney, R.L., & Hall, G.R. (1998). Diffrential equations. Pacific Grove: Brooks/Cole – Thomson Learning Inc.