If we are given a non-hom. DE. ay''+by'+cy=f. (they are positive constants)
1) Prove that all solutions approach f/c as x approaches infinity?
2) If c=0 what will happen, how will this change the solutions??
3) If c=0 and b=0 what will happen?
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If we are given a non-hom. DE. ay''+by'+cy=f. (they are positive constants) 1) Prove that all...
1a). Consider the equation ay" + by' + cy = d where d ∈ R and a, b and c are positive constants. Show that any solution of this equation approaches d/c as x → +∞. That is, given y(x) a solution we have lim y(x) as x → +∞ = d/c . 1b.) What happens if c = 0? 1c.) What about the case where b = c = 0?
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1. In class we showed that the function f : R → R given by (if>o 0 if a S0 was smooth (but not real analytic). Note that f(x) approaches a horizontal asymptote (y = 1) as a goes to positive infinity. (a) Show that f(x)+f(1-2)メ0 for all x E R, so that g : R → R given by g(x)- 70 is also a smooth function. (b) Prove that if 0 ifx-1. (c) Note...
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(1 point In general for a non-homogeneous problem y' + p(x) +(z) = f() assume that y. is a fundamental set of solutions for the homogeneous problemy" p(x) + (2) 0. Then the formula for the particular solution using the method of variation of parameters is where (z)/ and ()/() where W() is the Wronskian given by the determinant W (2) (2) W2) 31(2)/(2) dr. NOTE When evaluating these indefinite integrals we take the W(2) So we have the de...
A.7. In section 2.3, we learn how to solve ODEs of the form y'+ Px)y -f(x). But if P(x) and f (x) are both constants, the ODE can also be solved by separation of variables. Suppose P(x) = a and f(x) = b, where a and b are non-zero constants. We then have the ODE y' ay -b. If y(0)- Co, solve the initial-value problem.
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1. In class we showed that the function f : R → R given by (if>o 0 if a S0 was smooth (but not real analytic). Note that f(x) approaches a horizontal asymptote (y = 1) as a goes to positive infinity. (a) Show that f(x)+f(1-2)メ0 for all x E R, so that g : R → R given by g(x)- 70 is also a...
9. An n × n matrix A is called nilpotent if for-one non, negalivew m, we have Ao, If A is a nilpotent matrix prov conider invertible matrix. To prove this tell me what is (1 + AY first the case where m2 and in this case show th This should help you to see how to prove the general n x n identity matrix). that 1+ As an Hin at (1+A)---A) case. (I is the
9. An n ×...
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2 between x x2 Determine the average rate of change of f(x) 1 and x...