In Problems 1 through 12 first solve the equation f(x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f(x). Then analyze the sign of f(x) to de- termine whether each critical point is stable or unstable, and construct the corresponding phase diagram for the differen- tial equation. Next, solve the differential equation explicitly for x(t) in terms of t. Finally, use either the exact solution or a computer-generated slope field to sketch typical solution curves for the given differential equation, and verify visually the stability of each critical point. 7. dx dt = (x - 2)²

Q: Q1. Find the general solution for the following differential equation: -y² (x²ex² - y²)dx + xydy=0

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Q: Construct the inverse matrix (if it exists) for the following matrices: [12 61 (а) А %— 4 2. 1 -2]…

A: We will find out the required solution.

Q: Q1. Find the general solution for the following differential equation: -y² (x²ex² - y²)dx + xydy=0

A:

Q: In each question, draw the phase diagram (i.e. the graph with stationary points and arrows),…

A: "Since you have posted a question with multisubparts, we will solve the first three subparts for…

Q: H3. Solve the following differential equations d²y dy 1) +4 +4y= x=²e-²x, dx² dx dy d²y dx²…

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Q: 15. Obtain the differential equation of the family of straight lines through the point (5, 2).…

A: The given point is (5,2).

Q: 1. Find the critical points of the following differential equation. Then sketch the phase diagram…

A: Step1:Finding the critical points of the given differential Equationdx/dt = x (x2−25) To get the…

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Q: 4. Solve this linear differential equation by using the appropriate integrating factor +y =x - 1+

A: A differential equation is an equation that involves derivatives or differential coefficients or…

Q: Q1. Find the general solution for the following differential equation: -y (x2e x - y2)dx + xydy = 0

A: To find- Find the general solution for the following differential equation: x2e-y2x2 - y2dx + xydy =…

Q: The indicated function y₁(x) is a solution of the given differential equation. Use reduction of…

A: Solution is uploaded below.

Q: Q1. Find the general solution for the following differential equation: -y² (x²ex² - y²)dx + xydy=0

A:

Q: Solve the following initial values problem  y'(t)+y/2=4, y(-1)=0 y(t)=? Answer should be in terms…

A: Answer is- y(t)= 8(1-e^(-(t+1)/2)).

Q: Find the implicit solution of the differential equation (1+sin(y))dx + (1+x cos(y))dy = 0.

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Q: Problem 4. Consider the autonomous differential equation x' = sin x. a. Does the differential…

A: '

Q: 10. xy dx - (x + y') dy 0 Ans. 3 p3 +log y = C

A: given D.E isx2ydx-x3+y3dy=0---1is of the form Mdx+Ndy=0where M=x2y;N=-x3+y3∂M∂y=x2  ; ∂N∂x=-3x2

Q: Verify that the indicated family of functions is a solution of the given differential equation.…

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Q: xy! √√x²+ x² + y² +4

A: Here we have to use substitution to find the solution of given differential equation.

Q: The indicated function y1(x) is a solution of the given differential equation. Use reduction of…

A:

Q: 7. Find the first derivatives of each of the following: 'a. y= 10x – 4x² + 6x – 12 b. g(x) = x'e* c.…

A: we have to find the first derivative of the following: (d) ft=4t3sect (e) log65x+13

Q: I1 x₂ = x3 = || Solve the following initial value problem: 2 -10 7 +-60-~-() -1 5, z (0) 0-10 9 d dt…

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Q: The indicated function y₁(x) is a solution of the given differential equation. Use reduction of…

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Q: Classify the following as either linear or nonlinear differential equations and as either initial or…

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Q: 4) Find a function that satisfies the differenti equation, df = f(x) = 1 dx 2+1 K-2 and the initial…

A: I have given the answer in the next step. Hope you understand that