1. Consider the following second order linear equation with initial conditions (IVP): e'y" + y – y = 0, y(0) = 1, y'(0) = -1 (a) Convert the above initial value problem to an equivalent first-order linear system by defining new dependent variables x1(t), x2(t). Don't forget the initial values. (b) Write the IVP in part (a) using matrix-vector notation as ' = AT, 7(0) = xồ, where 7 (t) = :

Q: y (4) + y" +y" = 0, where y(0) = y'(0) = y″(0) = 0, y″(0) = 1

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Q: Obtain two linearly independent solutions valid near x = 0 x2y" + x(x-1)y' + (1-x)y = 0

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Q: Find two linearly independent solutions of 2x²y" - xy' + (−3x + 1)y = 0, x > 0 = x¹¹ (1 + α₁x + a₂x²…

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Q: If y, and y₂ are linearly independent solutions of t^2y" + 5y' +(2+t)y = 0 and if W(y₁, y₂)(1) = 3,…

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Q: onsider the following equation system: (P =r – Q – I+w P= α + Q | I = BQ + Vw here Q,P,I are the…

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Q: 3. Is y(x) = C,sin2x + C2cos2x, where C1 and C2are arbitrary constants, a solution of y" + 4y = 0?

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Q: 10. Consider the linear regression model with assumptions (i) to (iii) and (iv*) with V = k? I, + V1…

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Q: а. у' +3у%38, у(0) %3D4 b. 1 xdx+dy= 0, y(0)=8 y c. senxdx+ ydy=0 y(0)= -2 d. y' +2xy2 = 0, y(0) = 3

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Q: 2. Use separation of variables to solve (a) y' = -2y, y(0) = 1 (b) 4y' = y², y(0) = 4 %3D (c) y' + y…

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Q: If y = 2² is one solution of x²y" + 2xy' − 6y = 0, find the other linearly independent solution.

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Q: If y₁ and y₂ are linearly independent solutions of ty" + 3y'+te^5t y = 0 and if W(y1 y2)(1)= 3, find…

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Q: b) y(4) + y(3) – 7y(2) – y' + 6y = 0. c) y(4) – y(3) – 7y(2) + y' + 6y = 0.

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Q: y" – 3y' + 2y = 2x² + 1

A: The Solution is given below

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.