(1) Find the first two terms of both the inner and outer solutions of d²y + (1 + x)² dy + y = 0 dx2 dx for 0 < € <<< 1 with the boundary conditions y(0) = 0 and y(1) = e¹/2. [20 marks]

Q: Express (x + 4/x)dy/dx - y = 1 in the form y = f(x) where the curves passes through the origin.…

A: Given differential equation : x+4xdydx - y = 1                    ...........( 1 ) Multiplying both…

Q: 2. Use separation of variables to solve the following equations: (a) (y² + 1)dy + (e² + x)dx = 0.

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Q: (d) What can you say about the graph of y = 4(x) for x>x*? (e) Verify that y = x – 1 is a solution…

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Q: 2. Solve for y in terms of X for the following equations: a) In(1- 2y) = x b) In(y- 1)- In 2 = x +…

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Q: Consider the equation dy dx = y - 4x x-y which is not separable. (a). Show that the above equation…

A: Step 1: Step 2: Step 3: Step 4:

Q: Given the IVP DE: dy/dx = (x-y)^(1/2) IC: y(2) = 1. Determine whether the Existence and Uniqueness…

A: The given initial value problem dydx=x-y12, y2=1. We have to determine whether the Existence and…

Q: 2(y +1)e" dx + 2(e" – 2y)dy = 0 -

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Q: 1. For each F(x, Y) = 0, find dy/dx for each of the following: (а) у - 6х + 7 0 (b) 3y+12x + 17 =0…

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Q: 19.1 please provide all necessary solutions and explanations.

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Q: Using the given substitutions, find the solutions of the following equations. dy x+y dx 1-х-у (а) z…

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Q: Show that the given function y = 5e²x + x is a solution of the IVP 2x subjected to the condition…

A: The given initial value problem dydx-2y=1-2x subject to the condition y0=5 We have to show that…

Q: dy - 6/7 = 3x dx and y(- 1) = - 3. Find the function y(x) satisfying The function y(x) satisfying dy…

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Q: [y] + √2x+3] dx = (2x+3)dy, y(-1) = O

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Q: Solve the following Euler-Cauchy type equation: 1 (x² D² + 3xD+ 1)y = - 3xD+ 1)y = (1 – x)² ' d…

A: Given equation is, x2D2+3xD+1y=1(1-x)2 Take, (x2D2+3xD+1)y=0 Then assume, y=xn,…

Q: 2. Find (x) which satisfies (4e y − 1)dx + e dy = 0. O p(x) = e¯² + 5e-4x O p(x) = 3e e-4x O p(x)…

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Q: 2. 3y² dy/dx-1=3x²+6ydy/dx, y(0) -1 and determine the interval of x in which the solution is…

A: To solve the given initial value problem:  3y2dxdy​−1=3x2+6ydxdy​ with the initial condition y(0) =…

Q: dy 2x2 if y(1) = 2 dx у-5 dy – xys if y(0) = 8 dx dy = (6e3× + 1)dx if y(0) = 0 %3|

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Q: 1 Find the complete solution of y" - 2y' + 2y =x + e* cos x.

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Q: Find a family of solutions for x2y dx +(x2-x)dy=0. Assume x>0

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Q: H.W. Solve 1- xdy - 3ydx = x*y"'dx 2- (2xy? - 2y)dx + (3x²y- 4x)dy = 0 3- ly*(x+ 1) + yldx + |2xy +…

A: Our guidelines we are supposed to answer only one question. Kindly repost other question as the next…

Q: 4. (1 + e) dx + e(1) dy x (1) dy = 0 [Answer : x + yev = c]

A: The given problem is to find the solution for the given differential equation. We can solve it by…

Q: Use the generalized eigenvectors of A to find a general solution to the system x'(1) Ax(t), where A…

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Q: 3) Solve. dy e*y dx = e-y + e-2x-y (2y + 2) dy – (4x³ + 6x) dx = 0, y(0) = –3. dy + y = f(x), y(0) =…

A: a) Given differential equation is exydydx=e-y+e-y.e-2xyeydy=1+e-2xexdxyeydy=(e-x+e-3x)dx Integrate…

Q: 3. (1−x³) d²y dx² DE -+ (sin x)y' − 3xy = ln x(+1) TYPE ORDER DEGREE LINEARITY

A: Need to write down the type, order, degree and linearity of the given differential equation.

Q: Find a family of solutions for x2y dx +(x2-x)dy=0. Assume x>0

A: Given differential equation is: x2y dx + (x2-x) dy=0x2y dx=(-x2+x) dy This given differential…