Could someone explain how these to get these phase portraits by hand with ẋ=y and ẏ=ax-x^2 especially for a=0 case where you have eigenvalues all equal to zero?
Homework Answers
When eigen values are zero then lambda = 0 , When x = y , y = ax - x^2
When a=0 , all these are converging towards zero and from there onwards it is diverging.
When a<0 , all these are converging towards zero in circular form and then it is diverging.
When a> 0 , all these are converging towards zero in nodal form and then it is diverging
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Could someone explain how these to get these phase portraits by
hand with ẋ=y and ẏ=ax-x^2...
2) Sketch the phase portrait of the system x' (t) = Ax (t) if (a) 5=...
2) Sketch the phase portrait of the system x' (t) = Ax (t) if (a) 5= [ 9), P=[7"}] (1) 5= [ • ? ], P=[} >>]
Consider the linear system of first order differential equations x' = Ax, where x = x(t),...
Consider the linear system of first order differential equations x' = Ax, where x = x(t), t > 0, and A has the eigenvalues and eigenvectors below. Sketch the phase portrait. Please label your axes. 11 = 5, V1 = 12 = 2, V2 = ()
(1 point) Find the solution of x2y" + 5xy + (4 + 3x)y = 0, x...
(1 point) Find the solution of x2y" + 5xy + (4 + 3x)y = 0, x > 0 of the form Yi = x" Ž Cpx”, n=0 where co = : 1. Enter r = Cn = n = 1,2,3,...
Extra Credit Question:[4+4=8 pts) If E [exp(aX)] exists for a given constant a, then show that...
Extra Credit Question:[4+4=8 pts) If E [exp(aX)] exists for a given constant a, then show that for to (a) exp(-at)P(x >t) <E (exp(aX)], if a > 0. (b) exp(-at)P(X <t) <E (exp(aX)], if a < 0.
please explain each step 5.21 Let X and Y be independent random variables with fae-ax, x>0...
please explain each step
5.21 Let X and Y be independent random variables with fae-ax, x>0 fx(x) = 10. otherwise and Be-Bt, x>0 fr(y) = 10 otherwise where a and B are assumed to be positive constants. Find the PDF of X + Y and treat the special case a = B separately.
f(x) = 2.10 Consider the following algorithm (known as Horner's rule) to evaluate -ax': poly =...
f(x) = 2.10 Consider the following algorithm (known as Horner's rule) to evaluate -ax': poly = 0; for( i=n; i>=0; i--) poly = x * poly + ai a. Show how the steps are performed by this algorithm for x = 3, f(x) = 4x + 8x + x + 2. b. Explain why this algorithm works. c. What is the running time of this algorithm?
(6) Show that the semicircle C = {(x,y) = R2 | + y2 = 1, y...
(6) Show that the semicircle C = {(x,y) = R2 | + y2 = 1, y > 0} is a 1-dimensional manifold with boundary and the hemisphere D= {(x, y, z) | 22 + y2 + z2 = 1, 2 > 0} is a 2-dimensional manifold with boundary. (7) Suppose X is an n-dimensional manifold with boundary. Let ax denote the set of points in the boundary of X. Show that ax is an (n-1)-dimensional manifold.
Consider the autonomous differential equation dy dt = = y(k - y), t> 0, k >...
Consider the autonomous differential equation dy dt = = y(k - y), t> 0, k > 0 (i) list the critical points (ii) sketch the phase line and classify the critical points according to their stability (iii) Determine where y is concave up and concave down (iv) sketch several solution curves in the ty-plane.
2.5.9. The random variable X has a cumulative distribution function for xo , for xsO ....
2.5.9. The random variable X has a cumulative distribution function for xo , for xsO . for r>0 F(x) = z? 1 +x2 Find the probability density function of X.
2.5.9. The random variable X has a cumulative distribution function for xo , for xsO ....
2.5.9. The random variable X has a cumulative distribution function for xo , for xsO . for r>0 F(x) = z? 1 +x2 Find the probability density function of X.
2) Sketch the phase portrait of the system x' (t) = Ax (t) if (a) 5= [ 9), P=[7"}] (1) 5= [ • ? ], P=[} >>]
Consider the linear system of first order differential equations x' = Ax, where x = x(t), t > 0, and A has the eigenvalues and eigenvectors below. Sketch the phase portrait. Please label your axes. 11 = 5, V1 = 12 = 2, V2 = ()
(1 point) Find the solution of x2y" + 5xy + (4 + 3x)y = 0, x > 0 of the form Yi = x" Ž Cpx”, n=0 where co = : 1. Enter r = Cn = n = 1,2,3,...
Extra Credit Question:[4+4=8 pts) If E [exp(aX)] exists for a given constant a, then show that for to (a) exp(-at)P(x >t) <E (exp(aX)], if a > 0. (b) exp(-at)P(X <t) <E (exp(aX)], if a < 0.
please explain each step
5.21 Let X and Y be independent random variables with fae-ax, x>0 fx(x) = 10. otherwise and Be-Bt, x>0 fr(y) = 10 otherwise where a and B are assumed to be positive constants. Find the PDF of X + Y and treat the special case a = B separately.
f(x) = 2.10 Consider the following algorithm (known as Horner's rule) to evaluate -ax': poly = 0; for( i=n; i>=0; i--) poly = x * poly + ai a. Show how the steps are performed by this algorithm for x = 3, f(x) = 4x + 8x + x + 2. b. Explain why this algorithm works. c. What is the running time of this algorithm?
(6) Show that the semicircle C = {(x,y) = R2 | + y2 = 1, y > 0} is a 1-dimensional manifold with boundary and the hemisphere D= {(x, y, z) | 22 + y2 + z2 = 1, 2 > 0} is a 2-dimensional manifold with boundary. (7) Suppose X is an n-dimensional manifold with boundary. Let ax denote the set of points in the boundary of X. Show that ax is an (n-1)-dimensional manifold.
Consider the autonomous differential equation dy dt = = y(k - y), t> 0, k > 0 (i) list the critical points (ii) sketch the phase line and classify the critical points according to their stability (iii) Determine where y is concave up and concave down (iv) sketch several solution curves in the ty-plane.
2.5.9. The random variable X has a cumulative distribution function for xo , for xsO . for r>0 F(x) = z? 1 +x2 Find the probability density function of X.
2.5.9. The random variable X has a cumulative distribution function for xo , for xsO . for r>0 F(x) = z? 1 +x2 Find the probability density function of X.
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