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Mathematics LibreTexts

4.1E: Exercises

This page is a draft and is under active development. 

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Exercise 4.1E.1

Tanks T1 and T2 contain 50 gallons and 100 gallons of salt solutions, respectively. A solution with 2 pounds of salt per gallon is pumped into T1 from an external source at 1 gal/min, and a solution with 3 pounds of salt per gallon is pumped into T2 from an external source at 2 gal/min. The solution from T1 is pumped into T2 at 3 gal/min, and the solution from T2 is pumped into T1 at 4 gal/min. T1 is drained at 2 gal/min and T2 is drained at 1 gal/min. Let Q1(t) and Q2(t) be the number of pounds of salt in T1 and T2, respectively, at time t>0. Derive a system of differential equations for Q1 and Q2. Assume that both mixtures are well stirred.

Exercise 4.1E.2

Two 500 gallon tanks T1 and T2 initially contain 100 gallons each of salt solution. A solution with 2 pounds of salt per gallon is pumped into T1 from an external source at 6 gal/min, and a solution with 1 pound of salt per gallon is pumped into T2 from an external source at 5 gal/min. The solution from T1 is pumped into T2 at 2 gal/min, and the solution from T2 is pumped into T1 at 1 gal/min. Both tanks are drained at 3 gal/min. Let Q1(t) and Q2(t) be the number of pounds of salt in T1 and T2, respectively, at time t>0. Derive a system of differential equations for Q1 and Q2 that's valid until a tank is about to overflow. Assume that both mixtures are well stirred.

Answer

Q1=(rate in)1(rate out)1 and Q2=(rate in)2(rate out)2 .

The volumes of the solutions in T1 and T2 are V1(t)=100+2t
and V2(t)=100+3t , respectively. T1 receives salt from the
external source at the rate of (2 lb/gal) × (6~gal/min)=12 lb/min , and from T2 at the rate of
(lb/gal in T2)× (1~gal/min) =1100+3tQ2 lb/min . Therefore, (A) (rate in)1=12+1100+3tQ2 . Solution leaves T1 at 5~gal/min,
since 3~gal/min are drained and 2~gal/min are pumped to T2 ; hence
(B) (rate out)1=( lb/gal in T1)×(5~gal/min) =1100+2tQ1×5=5100+2tQ1 . Now (A) and (B)
imply that (C) Q1=125100+2tQ1+1100+3tQ2 .

T2 receives salt from the external source at the rate of (1 lb/gal) × (5~gal/min)= 5 lb/min , and from T1
at the rate of \) \mbox{(lb/gal in }T_1)\times\mbox{ (2~gal/min)
}=\displaystyle{1\over100+2t}Q_1\times2=\displaystyle{1\over50+t}Q_1 \mbox{ lb/min}\) .
Therefore, (D) (rate in)2=5+150+tQ1 . Solution
leaves T2 at 4~gal/min, since 3~gal/min are drained and 1~gal/min
is pumped to T1 ; hence (E) (rate out)2=( lb/gal in T2)×(4~gal/min) =1100+3tQ2×4=4100+3tQ2 . Now (D) and (E)
imply that (F) Q2=5+150+tQ14100+3tQ2 . Now
(C) and (F) form the desired system.

 

Exercise 4.1E.3

A mass m1 is suspended from a rigid support on a spring S1 with spring constant k1 and damping constant c1. A second mass m2 is suspended from the first on a spring S2 with spring constant k2 and damping constant c2, and a third mass m3 is suspended from the second on a spring S3 with spring constant k3 and damping constant c3. Let y1=y1(t), y2=y2(t), and y3=y3(t) be the displacements of the three masses from their equilibrium positions at time t, measured positive upward. Derive a system of differential equations for y1, y2 and y3, assuming that the masses of the springs are negligible and that vertical external forces F1, F2, and F3 also act on the masses.

Exercise 4.1E.4

Let X=xi+yj+zk be the position vector of an object with mass m, expressed in terms of a rectangular coordinate system with origin at Earth's center (Figure (4.1.3)). Derive a system of differential equations for x, y, and z, assuming that the object moves under Earth's gravitational force (given by Newton's law of gravitation, as in Example (4.1.3) ) and a resistive force proportional to the speed of the object. Let α be the constant of proportionality.

Answer

mX=αXmgR2XX3 ;
see Example~4.1.3.

Exercise 4.1E.5

Rewrite the given system as a first order system.

(a)x=f(t,x,y,y)y=g(t,y,y)

(b)u=f(t,u,v,v,w)v=g(t,u,v,v,w)w=h(t,u,v,v,w,w)

(c) y=f(t,y,y,y)

(d) y(4)=f(t,y)

(e) x=f(t,x,y)y=g(t,x,y)

Exercise 4.1E.6

Rewrite the system Equation (4.1.14) of differential equations derived in Example (4.1.3) as a first order system.

Exercise 4.1E.7

Formulate a version of Euler's method (Section 3.1) for the numerical solution of the initial value problem

y1=g1(t,y1,y2),y1(t0)=y10,y2=g2(t,y1,y2),y2(t0)=y20,

on an interval [t0,b].

Exercise 4.1E.8

Formulate a version of the improved Euler method (Section 3.2) for the numerical solution of the initial value problem

y1=g1(t,y1,y2),y1(t0)=y10,y2=g2(t,y1,y2),y2(t0)=y20,

on an interval [t0,b].

Answer

I1i=g1(ti,y1i,y2i),J1i=g2(ti,y1i,y2i),I2i=g1(ti+h,y1i+hI1i,y2i+hJ1i),J2i=g2(ti+h,y1i+hI1i,y2i+hJ1i),y1,i+1=y1i+h2(I1i+I2i),y2,i+1=y2i+h2(J1i+J2i).


This page titled 4.1E: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by William F. Trench.

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