Use the Laplace transform to solve the given initial-value problem. y'' + 2y' + y = 0,  y(0) = 1,  y'(0) = 1 y(t)=

Q: y''+y'=0 where y(0)=0, y'(0)=1 laplace transform of derivatives

A: Given that:-  y" + y' = 0 and y(0)= 0, y'(0) =1 We find laplace transform of this differential…

Q: Find the general solution of the following first-order ODE xy' = y + 2x3 sin?

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Q: Find f'(z) of f(z) = 3πi.

A: Compare the given function f(z) = 3πi. with f(z)=u(x,y)+i v(x,y) So,  u(x,y) =0 and v(x,y) = 3π

Q: Find y as a function of x if x'y" + 5xy' + 4y = 0, y(1) = 9, y'(1) = -2. y =

A: Solution :-

Q: Find y as a function of a if x?y" + 15xy + 49y = 0, |3D y(1) = -5, y'(1) = 4. y =

A: Given differential equation is  x2y''+15xy'+49y=0 given y(1)=-5,y'(1)=4

Q: Find y as a function of a if y(1) = 3, y'(1) = -7. y = r²y" + 3xy' - 63y = 0,

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Q: Solve the ODE  y'+(1/x)y-ex=0, when x>0

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Q: let h(y)=(y-3y2)5(y2+2)6  h'(y) in simplified form.

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Q: Find a function y(æ) such that byy' = x and y(6) = 5. y =

A: y' = dy/dx y(6) = 5 means at x=6 => y=5

Q: If t²y" – ty – y = 0 ,y(1) = 3 and y'(1) = -1, then y(4) = . . . %3D

A: Given differential equation, t2y''-12ty'-y=0,y1=3,y'1=-1 Find the value of y4.

Q: Find y" * y" at (1,1) of x'y =1

A: We have to find derivative

Q: Solve The ODE  y'=2y

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Q: Find y as a function of if y(0) = 19, y'(0) = 19, y"(0) = 4, y"(0) = 0. y(x) = y(4) 4y" + 4y = 0,

A: y(4)−4y′′′+4y′′=0y(0)=19,y′(0)=19,y′′(0)=4,y′′′(0)=0

Q: f(z) 2+3x1x², find f'(5).

A: f(x)=2+3x−4x2, find f′(5).

Q: y'' + 4y = cos2t y(0) = -2 and y'(0) = 0 Solve the IVP using the Laplace Transform.

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Q: y''-3y'+10y=e-2t , y(0) =0, y'(0)=1

A: We have to solve given problem:

Q: by s- differentiation find L(te-3t) = 1 .aO (s- 3) 1 .b O (s-3)2 (s+3) .dO (s+3) 2 1.

A: The solution is given as

Q: Obtain general solution of ODE y''-y=exp(x) + cos(2x)

A: Step 1. Homogeneous Solution (y_h):We start by assuming the solution for the homogeneous equation…

Q: Sin (7€) + Cos (SE) 5지

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Q: y" (f) + (1 – t) y' (4) + e'y (t) = – 5t; y (0) = - 3; y' (0) = 4 into an in- tegral equation.…

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Q: For the Cauchy-Euler equation x2y''+xy'+y=x2 if you substitute z=ln(x), y'=(1/x)(dy/dz), and…

A: The Cauchy-Euler equation x2y''+xy'+y=x2. Find the value of a+b+c=?

Q: (e) Compute y' if y2 = xe + ye-

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Q: Find y as a function of x if x'y" – 9xy' + 25y = x", y(1) = 2, y'(1) = -5. y =

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Q: Find y as a function of a if a?y" + 13ry' + 36y = a, %3D y(1) = -2, y'(1) = -1. y =

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Q: Find the general solution of the following first-order ODE xy' = y + 2x' sin² (2)

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