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Calculus and Analysis

Solution to Nonlinear Differential Equation

Please see the attached file for the fully formatted problems. 1. Consider the nonlinear differential equation attached a. Find the solution to this differential equation satisfying y(0) = y0 where y0 does not equal +/- 1. What is the solution if y0 = +/-1. b. What happens to the solution as t--> infinity for y0 > -1?

First Order Differential Equation

Problem: First find the general solution of the linear ODE in each IVP by following the steps of the procedure. Then use the initial condition to find the solution of the IVP. Discuss that solution's qualitative behaviour as t --> +(SYMBOL). Give the largest t-interval on which the solution is defined: y' + 2y = 3, y(0) = 1

Ode (Boundary Condition, Implicit and Explicit Solutions)

Consider the ODE y' = y2/x subject to the boundary condition y(1)=1. Find an implicit solution of the form H(x,y) = constant, then find an explicit solution of the form y=y(x). What is the largest x-interval on which the solution is defined? *(Please see attachment for proper citation of symbols and numbers)

Differential Equations

For each of the following ordinairy differential equations, indicate its order, whether it is linear or nonlinear, and whether it is autonomous or non-autonomous. a) df/dx +f^2=0 (See attachment for all questions)

Multiple integrals

Consider the vector field F(x,y)= (-yi+xj)/(x^2+y^2) Question1)Show that F is the gradient of the polar angle function teta(x,y)=arctan(y/x) defined over the right half-plane x>0 . Question2)Suppose that C is a smooth curve in the right half-plane x>0 joining two points : A:(x1,y1) and B(x2,y2).Express "integral(F.dr)"on

Integral Equation Solved

Please show me how to solve this equation - can it be solved by substitution or am I on the wrong track? *(Please see attachment for equation)

Instantaneous rate of change: Pool problem

A swimming pool is 20 ft wide, 40 ft long, 3 ft deep at the shallow end, and 9 ft deep at its deepest point. If the pool is being filled at a rate of 0.8 ft^3/min, how fast is the water level rising when the depth at the deepest point is 5 ft?

Rate of change

A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has height 50 cm. If the trough is being filled with water at a rate of 0.2 m^3/min, how fast is the water level rising when the water is 30 cm deep?

Instantaneous Rate of Change of Water

A trough is 10 ft long and its ends have the shape of isosceles triangles that are 3 ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of 12 ft^3/min, how fast is the water level rising when the water is 6 inches deep?

Function and Differential Equations

Please assist me with the attached problems relating to functions and (partial) differential equations. Thanks very much. Please tell me which is true and which is false. Then please explain why it is true or false. As you see c, d, and e are the answers I believe and next to some in parenthesis are some basic explanations.

2nd order ODE

64y'' + 64y' +18y = 0 y(0)=4 y'(0)=4 find y as a function of t

Multivariable Calculus: Fastest Decreasing Function

Question 1) Find the direction from (-3, 1, 2), in which g(x,y,z)=x^2*y*z-2*z^3 decreases fastest. question 2) Follow the line in the direction you found in part 1) to estimate, using linear approximation, the location of the point closest to the coordinates (-3,1,2) at which g=1;Do not use a calculator; Express the answer us

Multivariable Calculus: Partial Derivative

Let x = (1/2)(u^2-v^2), y=uv, and f=f(x,y) [QUESTION 1] use the chain rule to derive the change of variables formula in matrix form: (fu,fv)=A*(fx,fy) {actually it is vertical , so fu is at the top and fv is at the bottom. Same for fx and fy: fx is under fy; sorry for the notation I cant do it another way} [QUESTION 2]

Multivariable calculus

Consider the function : f(x,y) = x(x-1)(x-2) + (y-1)(x-y) [QUESTION 1]find the maximum and the minim values of the directional derivative (df/ds)]u at ( 1 , 3/2 ) as u varies . ( (df/fs)]u : I can't write the symbol clearly but it means : the derivative of f according to s on the directi

Multivariable calculus

Consider a triangle in the plane, with angles , a, b , c. Assume that the radius of its circle is equal to 1. 1) by decomposing the triangle into six right triangles having the incenter as a common vertex, express the area A of the triangle in term of a, b , c ( the answer should be a symmetric expression). Then use the resu

Explicit/implicit solutions

Find an explicit or implicit solutions to the differential equation: (x^2 + 4xy)dx + xdy = 0 "F(x,y) such that the solutions are F(x,y)=c for an arbitrary constant c".