1. Use the Chain Rule to find u and ou for w = x²y3 where x = dw at s2 +t2 and y = ds

Q: Solve the following initial value problem using Laplace transforms. y"+2y'-3y=3e¹, y(0) = 0,…

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Q: Use the method of Laplace transforms to solve the symbolic initial value problem y" + 9y = 38(t –…

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Q: Use the Laplace transform to solve the initial value problem. (a) y" – y' – 6y = 2, y(0) = 1, y'(0)…

A: Laplace transform

Q: By using Laplace transform to solve the IVP: y" - 4y' +9y=t, with y(0) = 0 and y'(0) = 1 Then Y(s)…

A: Given IVP is  y"-4y'+9y=t ......(1) y(0)=0,y'(0)=1 .....(2) We fin Y(s).

Q: EXAMPLE 2: dz 1 Given that z = x+y,where x = and y 2r. Find by: dt 1 a. Substituting x= and y = 2r'…

A: From the given problem:   z=x2+y2 ; x=1t2 ; y=2t3 a). Now, substitute x=1t2 ; y=2t3 in z=x2+y2 .…

Q: dw Use an appropriate form of the chain rule to find dt w = In(10x² – 4y + 5z³); c = t', y = tỉ, z =…

A: From the given function: w=ln10x2-4y+5z3 ; x=t12 ; y=t23 ; z=t-2 As we know that: The Power law:…

Q: 5. Use Laplace transforms to solve the initial value problem y" + 3y - 10y = 13(1-cos(t)), y(0) = 1,…

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Q: 4x Find the linearization L(x,y) of the function f(x,y) = e cos (3y) at the points (0,0) and0,

A: Answer is given below

Q: 3. Given w = x² + y² + z²; z = te"+s, use the CHAIN RULE to find x = ret+s; y = se**; and aw а. ar…

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Q: Solve the following Initial Value Problem using Laplace Transform y" – 3y' + 2y = e¬4t, y(0) = 1,…

A: Consider the given information. y''-3y'+2y=e-4t, y0=1, y'0=1 Now, apply the Laplace transformation.…

Q: 5.) Suppose z = f (x, y) , where x=x(t,r) and y= y(t,r). Assuming all the functions involved have…

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Q: Q1) Suppose w = x = cos? t, y = sin? t, z = dw Evaluate at t = 3, by using the Chain Rule dt

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Q: b) Use Laplace transforms to solve the following initial value problem; d²y dy 6- dt + 9y = t²e3t,…

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Q: use the Laplace transform to solve the given initial-value problem. Q. y′′+y =6cos2 t, y(0)=0,…

A: Given differential equation is And initial condition is

Q: y" + 3y/ + 2y = e-t, y(0) = 0, y'(0) = 2.

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Q: (a) express dw/dt as a function of t, both by using the Chain Rule and by expressing w in terms of t…

A: Express dw/dt as a function of t by using the Chain Rule. Use the sum rule and split the bracket.…

Q: Suppose z = x² sin y, x = 2s2 + 3t2, y = 4st. dz and A. Use the chain rule to find ds as functions…

A: Given: z=x2siny x=2s2+3t2 y=4st To find: Using chain rule, ∂z∂s and ∂z∂t ∂z∂s1,-2 and ∂z∂t1,-2

Q: Suppose w = In(8x² + 2y² + 3z²), ¤= 9 cos(t) , y = 9 sin(t) , and z = 8Vt. Compute dw/dt in two…

A: The partial derivative of a multivariable function with respect to any of the variables can be…

Q: d2y/dt2-y=t2 (laplace transforms)

A: To solve the given differential equation using Laplace transform. Here we will use the following…

Q: Use the Laplace transform to solve the initial value problem. y" – y' – 6y = 2, y(0) = 1, y'(0) = 0…

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Q: 10. Use the method of Laplace transform to solve the initial value problem y" – 2ay + (a? + B²)y =…

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Q: Use the Laplace transform to solve the initial value problem y" + y' – 2y = e# with y(0) =1 and…

A: Given differential equation :

Q: In this question we work with the ODE dy +2ty = –T cos (T t) dt and initial condition y (0) = 0.

A: Given differential equation is, 1+t2dydt+2ty=-πcosπt The equation is rewritten as,…

Q: y" + 4y = 8t, y(0) = 2, y'(0) = 0.

A: Here we will use Laplace transform to solve the Initial value problem.

Q: Suppose that w = f(x, y, z), x = h(s), y= g(s, t), z= k(s,t) and all the conditions required for the…

A: By using the chain rule for partial differentiation we will find the value of it.

Q: az and as az Use the chain rule to find at z = In(4x + 3y), x = s sin(t), y = t cos(s) az as az at

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Q: (a) express dw/dt as a function of t, both by using the Chain Rule and by expressing w in terms of t…

A: We find dw/dt using chain rule. Here plug x = cos t, y = sin t, z = 4sqrt(t) and simplify.

Q: 27. Consider the piecewise path which goes along (t,3 – 3t, -t) for 0 <t<1, then along (1,t,-1+ t)…

A: Line Integral along the given parametric curve.

Q: a When reduced to a linear DE of order one in z, the Bernoulli's equation dx becomes dz A + z( − 1)…

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Q: Use the chain rule. If W=u^3 and U=t-1/t+1, find dw/dt when t=1

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Q: 19. Let f(x, y, z) = + sin(3z) + e"-2". Find the rate at which f is changing at (2, 4,0) in the…

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Q: 2. Use the Chain Rule to find for z = e¤/y where x = st and y = s2 +t

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Q: 8. Use the method of Laplace transform to solve the initial value problem y" – 20y' + a'y = (a –…

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Q: Suppose z = x´ sin y, x = 2s2 + 3t2, y = 6st. A. Use the chain rule to find z and as functions of x,…

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Q: dz a²z and 1 a) Find- Usin chain rule. If z = e*y – Sin xy, x=-, y=t? дх ду dt

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Q: 6. Solve the second order linear IVP d²y Y at? dy 2 = 1. "6)-1 dt (dt Hint: dt- dy substitute dy…

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Q: y" – 2y' = 6e*, y(0) = 1, y'(0) = 2.

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Q: Solve the following initial value problems using Laplace Transforms: d^2y/dt^2 + 4 dy/dt = cos(t-3)…

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Q: Consider the following. w = xy2 + x2z + yz², x = t², y = 5t, z = 5 (a) Find dw/dt using the…

A: Details calculation is given below

Q: If y, (x) = e** is a solution of the D.E y" + p(x)y' + q(x)y = 0, where p(x) and q(x) are continuous…

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Q: Show that the solution y(t) of the initial value problem has the Laplace Transform y" + 2y' - 3y = 6…

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Q: Consider the following. w = xyz, x = s + 5t, y = s – 5t, z = st? (a) Find aw/as and ôw/at by using…

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Q: If z = f(x,y) = tan x =t cose and y =t sin@. Use the Chain Rule method to find y as functions of t…

A: Given that z=f(x,y)=tan-1xy Where x=tcosθ,y=tsinθ Substituting, the values,…

Q: Use Laplace transforms to solve the initial-value problem: y" (t) – y'(1) + 1 = 0, subject to y(0) =…

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Q: Əw/əs and əw/ət by using the appropriate Chain Rule. w = x cos yz, x = s², y = t², z = s - 5t

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Q: Use Laplace transforms to solve the following initial value problem. y (t) = 4 y, y(0) = 1.

A: Laplace taking first we are getting L(y) from that we can get our required solution y.

Q: Solve the following Initial Value Problem using Laplace transform: y′′−7y′+6y=e−9t ;y(0)=1,y′(0)=1

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Q: Suppose z = x² sin y, x = -s² – 21² , y = 8st. dz A. Use the chain rule to find and * as functions…

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Q: Suppose z = 2xy where x = t^2 + 1 and y = 3 - t. Use the chain rule to determine which of the…

A: In the given question the functions are given as z=2xy , x=t2+1 , y=3-t . The objective is to find…

Q: Use the chain rule to find ∂w/∂t given w =√ x^2 + y^2 + z^2 and x = cos(st), y = −2s ln(t) and z = s…

A: Solution:Given      w=x2+y2+z2x = cos(st), y = −2s ln(t) and z = s − 3t.To find…