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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).
25.
[Hint: The resulting differential equation for r is a Bernoulli differential equation. See Section 2.5.]
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Differential Equations with Boundary-Value Problems (MindTap Course List)
- Example 11.1. Classify the following equations : a²u 22u d²u du du +4. +4 J + 2- = 0 Əxdy oy² ax dy a²u •+ (1 - y²¹). = 0, -∞ < x ∞, -1arrow_forward2. Which of the following is a general solution to the following: x²y" + xy' + (36x² - 1) y (Hint: As discussed in the lecture, use Y, only when J, and J-, are linearly dependent). A. y = c₁J₁(2x) + C₂J_1(2x) 6 B. y = C₁J₁(x) + C₂Y₁(x) 3 3 C. y = c₁₂/₁(6x) + C₂Y₁(6x) 0 D. y = c₁J₁(6x) + c₂] _1(6x) 2arrow_forward4. Obtain the differential equation of the family of lines passing through the center of the conics described by the equation x2 + 4y² + 2x – 8y +1 = 0. -arrow_forwardMatch each linear system with one of the phase plane direction fields. (The blue lines are the arrow shafts, and the black dots are the arrow tips.) ? ✓ | 1. z ' = || a' ? 2. ': = ? 3.' = 4. a: = 11 8] -10 3 1 5 -2 1 -5 -13 10] -10 x2 A x2 с x1 (x2 B 2x2/ D Note: To solve this problem, you only need to compute eigenvalues. In fact, it is enough to just compute whether the eigenvalues are real or complex and positive or negative.arrow_forward13. The standard form of 3dy-xydx = 3y3e4x/3 by Bernoulli's equation is: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_forward15 Give the properties for the equation 3y 2 - 2y + x + 1 = 0. Center (-1, 1/3) (-2/3, 1/3) (-8/9, 1/3)arrow_forwardty'''+2y''+y'+ty=0 What is the Wronskian of linearly independent solutions?arrow_forwardProblem 5. Find and Classify the critical point of (x,y)=192x³+y²–4xy² on the triangle with vertices (0,0), (4,2) and (-2,2).arrow_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_forward5. Given the system X" - ()* 10 2 (a) Find the general solution. (b) Find the solution that satisfies the initial condition (x(0), y(0)) = (0, –7). Report your solution as one vector.arrow_forwardQ1: Formulate the equation of plane in space that contain two points (2,4,-1) and (1,3,-2) and perpendicularly intersects with the plane 3x+5y+4z=487. Q2: Solve the following differential equation in two ways. (2xy + x?)dx + (x2 + y²)dy = 0 Q3: Find the shortest line from a point to a plane, justify your answer by calculations. Hint: you can choose any coordinates of the points and equation of the plane. Q4: Design the biggest box (volume) without cover that made from 6m2 of aluminum. Hint: Use Lagrange Multipliers Methodarrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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