For Problems 1-7, show that the given functions are solutions of the system
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Differential Equations and Linear Algebra (4th Edition)
- Consider the system of linear equations in x and y. ax+by=ecx+dy=f Under what conditions will the system have exactly one solution?arrow_forwardWhat operations can be performed on a linear system to arrive at an equivalent system?arrow_forwardSolve for x and y in the matrix equation 2AB=I, given A=[1123] and B=[x2y5].arrow_forward
- Use Cramers Rule to solve the system of linear equations for x and y. kx+(1k)y=1(1k)x+ky=3 For what values of k will the system be inconsistent?arrow_forwardSolve for x,y,z and w in the matrix equation wxyx=-432-1+2ywzx.arrow_forwardGiven (1 − x2)y′′ − 2xy′ + 12y = 0 give two linearly independent solutions. Give at least 4 non-zero terms for each solution, UNLESS one of the solution is a polynomial, in which case, give the polynomial.arrow_forward
- Please answer all questionsarrow_forwardFind the general solution of the following linear system: [1] 4 3 y' = yarrow_forwardConsider the two tanks shown in the figure below. Assume that tank A contains 50 gallons of water in which 25 pounds of salt is dissolved. Suppose tank B contains 50 gallons of pure water. Liquid is pumped into and out of the tanks as indicated in the figure; the mixture exchanged between the two tanks and the liquid pumped out of tank B are assumed to be well stirred. We wish to construct a mathematical model that describes the number of pounds x₁(t) and x₂(t) of salt in tanks A and B, respectively, at time t. dx₁1 dt dx1 dt pure water 3 gal/min mixture 4 gal/min This system is described by the system of equations 1 50 2 25 dx2 dt 2 25 2 25*1 1 + -X2 mixture 1 gal/min -X2 B with initial conditions x₁(0) = 25, x₂(0) = 0 (see (3) and the surrounding discussion on mixtures on page 107). What is the system of differential equations if, instead of pure water, a brine solution containing 3 pounds of salt per gallon is pumped into tank A? -2 25*1 + 50x2+6× dx2 2 - (25)*₁- (25)×₂2 - = x2 dt…arrow_forward
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