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In Problems 23–26 solve the given nonlinear plane autonomous system by changing to polar coordinates. Describe the geometric behavior of the solution that satisfies the given initial condition(s).
26.
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Chapter 10 Solutions
Differential Equations with Boundary-Value Problems (MindTap Course List)
- (3.3) Find the fixed points of the following dynamical system: -+v +v, v= 0+v? +1, and examine their stability.arrow_forward6. Consider the dynamical system dx - = x (x² − 4x) - dt where X a parameter. Determine the fixed points and their nature (i.e. stable or unstable) and draw the bifurcation diagram.arrow_forwardThis is the fourth part of a four-part problem. If the given solutions ÿ₁ (t) = - [²2²], 2(0)-[¹7¹]. form a fundamental set (i.e., linearly independent set) of solutions for the initial value problem 21-² -21-2 y' = - [22 12t¹+2t 27 2t ¹-2t 2 [²], 2t | Ü‚ ÿ(5) — [34], t = t> 0, impose the given initial condition and find the unique solution to the initial value problem for t> 0. If the given solutions do not form a fundamental set, enter NONE in all of the answer blanks y(t) = (arrow_forward
- Suppose that we wish to find the rest solution of the equation y" + 2y' + 10y = 10arrow_forwardProblem 2. Consider the equation: x?y"(x) – xy' +y = 0. Given that yı(x) = x is a solution of this equation. Use the method of reduction of order, find the second solution y2(x) of the equation so that y1 and y2 are linearly independent. (Hint: y2(x) should be given in the form y2(x) = u(x)y1(x). Substitute it into the equation to find u(x).) %3Darrow_forward2.5 Consider the system x' = (² 9) x. Find the special values of a for which the phase plane changes type. (Assume det A=0).arrow_forward
- 8.1 I only need number 22 pleasearrow_forward4. Find the equilibria of the autonomous system of equations, classify them and x=2-2√1+x+y = exp(-x+2y+ y²)-1' sketch the phase portraits in their neighborhoodsarrow_forward1. Find the complete solution of each of the following: 3y" + y' + 5y = 0 b. y"+5y" + 3y'-9y 0 c. (D2- 25)2 (D2- 4D+ 13}3 y = 0 a. y"- %3D %3D d. (D4 + 4D2) y = 0 %3D e. (D3 D2 + 9D-9) y= 0arrow_forward
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