Given Y = c1eλ1tx1 + c2eλ2tx2 +· · ·+cneλntxn is the solution to the initial value problem: Y = AY, Y(0) = Y0 (a) show that Y0 = c1x1 + c2x2 +· · ·+cnxn (b) let X = (x1, . . . , xn) and c = (c1, . . . , cn)T. Assuming that the vectors x1, . . . , xn are linearly independent, show that c = X−1Y0.

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A: To evaluate the following integral.

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Q: Determine general solution of: y(5) - 8y" = t² + 6t+4+et

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Q: (х? + 2у — 10у3х) if z у? (у? + 4х + 5у) dz -, find || дх

A: Given:

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Q: If v = -2 4 and w = 4 11 3 " compute proj(w) and proj(w).

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Q: 3. y = (x+b)(x-a). (x-b) ; y

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Q: If z=x2sinh 4 then 02z дхду

A: Explanation of the answer is as follows

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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.