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Chapter 9: Q6E (page 442)

Solve the differential equationf'(t)-2f(t)=e2tand find the solution of the differential equation.

Short Answer

Expert verified

The solution is ft=te2t+Ce2t.

Step by step solution

01

Definition of first order linear differential equation

Consider the differential equation f'(t)-af(t)=g(t)whereg(t)is a smooth function and 'a' is a constant. Then the general solution will be f(t)=eat鈭�e-atg(t)dt.

02

Determination of the solution

Consider the differential equation as follows.

f't-2ft=e2t

Now, the differential equation is in the form as follows.

f't-aft=gt, where g (t) is a smooth function, then the general solution will be as follows.

role="math" localid="1660799880430" ft=eat鈭�e-atgtdt

03

Compute the calculation of the solution.

Substitute the value e2tforgtand2forainft=eat鈭�e-atgtdtas follows.

ft=eat鈭�e-atgtdtft=e2t鈭�e-2txe2txdtft=e2t鈭�e0tdtft=e2t鈭�tdt

Simplify further as follows.

ft=e2t鈭�tdtft=e2tt+Cft=e2tt+e2tC

Hence, the solution for the linear differential equation f't-2ft=e2tisft=te2t+Ce2t.

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Most popular questions from this chapter

Question: Solve the system

dx鈫�dt=Ax鈫�

for a 2x2 matrix A with complex eigenvaluesp卤iq .

1. Finde2蟺i .

Solve the initial value problem inf''(t)鈥�9f(t)=0;f(0)=0,f'(0)=1

Consider a linear system dx鈫�dt=Ax鈫�of arbitrary size. Suppose x1(t)and x2(t)are a solution of the system. Is the sumx鈫�(t)=x鈫�1(t)+x鈫�2(t) a solution as well? How do you know?

Consider a quadratic formq(x鈫�)=x鈫�路Ax鈫�of two variables. Consider the system of differential equationsdx1dt=鈭�q鈭�x1dx2dt=鈭�q鈭�x2

Or more sufficiently

dx鈫�dt=gradq

  1. Show that the systemdx鈫�dt=gradq is linear by finding a matrix B(in terms of the symmetric matrix A) such thatgradq=Bx鈫�
  2. When q is negative definite,draw a sketch showing possible level curves of q. On the same sketch draw a few trajectories of the system localid="1662089983392">dx鈫�dt=gradq.What does your sketch suggest about the stability of the systemlocalid="1662089999708">dx鈫�dt=gradq
  3. Do the same as in part b for an indefinite quadratic form
  4. Explain the relationship between the definiteness of the form q and the stability of the systemlocalid="1662090018926">dx鈫�dt=gradq

The temperature of a hot cup of coffee can be modelled by the DE

T'(t)=-k(T(t)-A)

(a) What is the significance of the constants K and A?

(b) Solve the DE for T (t) in terms of K, A and the initial temperatureT0
See all solutions

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