Help with my diffeque homework Solve the following linear equation (t2 + 1) dy /

Question

Help with my diffeque homework

Solve the following linear equation (t2 + 1) dy / dt + 3ty = 6t, y(0) = 3. more 2.1: 8, 10, 15 Find the general solution of y(4) + 2y" + y = 0. more 4.2: 14, 16, 20, 28, Solve 3x2y + 2xy + y3 + (x2 + y2)dy / dx = 0. more 2.7:7, 9 28, 29 Solve the equation t2y' + 2ty - y3 = 0, t > 0. More 2.4: 29, 30 Use the method of variation of parameters to find the general solution of the differential equation y" + 4y' + 4y = t-2e-2t for t > 0. more 3.6: 4, 10, 14, 17 Determine the general solution of y" + y = 3sin(t) + t cos(2t) by the method of undetermined coefficients. for high order ODE, do 4.3: 9, 11, 13

Explanation / Answer

1) Divide on both sides by (t^2 + 1) . It becomes of the form y' + p(t) * y = q(t). Solution to this is

y* e ^ (integral p(t) dt) = integral ( q * e^(integral p(t) dt))

2) put y = e ^ mx . we get, m^4 + 2m^2 + 1 =0 .Therefore m is i,i,-i,-i . e^ix = cos (x) + i sin (x) .

Therefore solutions are , cos(x) , sin(x). As they are repeated roots, xsin(x) , xcos(x) are also solutions.

4) put y = t^m. we get the solution. this is known as a cauchy-euler form.

6)let y = asint + btsint + ccost + dtcost . Substitute and compare the coefficients...

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