Give the general solution of the differential equation y” +...

Give the general solution of the differential equation y'' + 25 y = -4 cos(5 x) + 3 sin(5 x) a) y = C_1 e^5 x + C_2 e^-5 x + 2/5 x sin(5 x) - 3/10 x cos(5 x) b) y = C_1 sin(5 x) + C_2 cos(5 x) - 2/5 x sin(5 x) + 3/10 cos(5 x) c) y = C_1 e^5 x + C_2 e^-5 x + 2/5 x sin(5 x) + 3/10 x cos(5 x) d) y = C_1 sin(5 x) + C_2 cos(5 x) - 2/5 x sin(5 x) - 3/10 cos(5 x) e) y = C_1 sin(5 x) + C_2 cos(5 x) - 2/5 x sin(5 x) - 3/10 x cos(5 x) f) None of the above.

Transcript

00:01 For the differential equation y double dash plus 25 pi equal to minus 4 cos 5x plus 3 sine 5x.

00:14 So to find its general solution to find its general solution general solution we first we first find this mark this equation 1 we first find the solution of solution of homogeneous homogeneous differential equation that is that is y double dash plus 25 y equal to 0.

00:48 So it is we first find the complementary solution.

00:52 So the characteristic the characteristic characteristic polynomial mark this equation 2 polynomial for differential equation 2 is for y double dash we have m square plus 25 equal to 0 and this implies m square equal to minus 25 this further implies m equal to square root plus minus minus 25 and which is equal to plus minus 5 iota.

01:25 So the complementary solution is the complementary solution is the complementary solution is say yc which is equal to yc of x ycx which is equal to c1 sine 5x plus c2 cos 5x.

01:50 Now we find the we find the particular solution of particular solution of differential equation 1.

02:04 So let us assume let assume we find the particular solution differential equation using using the method of the method of undetermined coefficient undetermined coefficient undetermined coefficient.

02:27 So let us assume the particular solution the particular solution is of the form solution is of the form is of the form ypx equal to a1x cos 5x plus a2x sine 5x.

02:52 So now find the first derivative it's so ypx equal to yp dash x equal to so its first derivative is we find it first derivative that is a1 into cos 5x plus a1x into the derivative of cos 5x is minus 5 sine 5x plus a2 sine 5x plus a2x into the derivative of sine x is cos 5x into 5.

03:31 So after simplifying we have a1 after simplifying we have a1 cos 5x minus a1x sine 5x plus a2 sine 5x plus a2x 5a2x here we are also 5 5a2x into cos 5x.

03:57 Now we find its second derivative.

04:00 So the second derivative of yp double dash x equal to minus 25.

04:06 So first we derivate it with respect to x second derivative is 5 into minus 5 sine 5x...

Question

Please give Ace some feedback