A) Find the general solution for the following second-order...

a) Find the general solution for the following second-order differential equations: i. d"y/dx" - 9y = 0 ii. 2 d"y/dx" + 16 dy/dx + 32y = 0 iii. d"y/dx" + 4 dy/dx + 5y = 0 b) Solve the following second-order differential equations: d"y/dx" + 10 dy/dx + 21y = 0, y(0) = 2, dy/dx = 2

Transcript

00:01 Hi, from the question given that here in part a we need to find the general solution of the given differential equation.

00:08 So, first d squared y by dx squared minus 9y is equal to 0 so that the auxiliary equation will be m squared minus 9 is equal to 0.

00:21 So, m squared is equal to 9.

00:23 So, m is equal to plus or minus 3.

00:26 So, that the general solution y of x is equal to c1 e to the power of 3x plus c2 e to the power of minus 3x.

00:35 Now, let us move on to the second one.

00:39 So, in the second one, this is first one, 2 times of d squared y by dx squared plus 16 dy by dx plus 32 y is equal to 0.

00:52 So, here is also first we need to find the auxiliary equation.

00:55 So, auxiliary equation can be written as 2m squared plus 16m plus 32 is equal to 0.

01:03 So, we need to factorize 64 as 8 8 z.

01:08 So, we have 2m squared plus 8m plus 8m plus 32 is equal to 0.

01:15 So, here we need to take 2m as a common.

01:18 So, m plus 4m not m.

01:23 So, we have already taken m here.

01:25 So, m plus 4 plus taken 8.

01:28 So, we have m plus 4.

01:30 So, that is equal to 0.

01:32 So, that the factors are 2m plus 8 times of m plus 4 which is equal to 0.

01:40 So, here the value of m is equal to minus 8 by 2 is minus 4 and m is equal to minus 4.

01:47 So, this is the factors.

01:48 Therefore, the general solution will be y of x is equal to ax plus b times e to the power of minus 4x.

02:00 And in the third part it is given that b squared y by dx squared plus 4 dy by dx plus 5y is equal to 0.

02:14 So, that the auxiliary equation is m squared plus 4m plus 5 is equal to 0.

02:21 So, when we factorize this m is equal to minus 4 plus or minus square root of b squared is 16 minus 4 times of a is 1 c is 5 divided by 2.

02:36 So, that is equal to minus 4 plus or minus square root of 16 minus 20 became minus 4 divided by 2.

02:43 So, that is equal to minus 4 plus or minus 2i divided by 2.

02:49 So, which is equal to minus 2 plus or minus i.

02:55 Therefore, the general solution will be y of x is equal to e power minus 2x times c1 cos x plus c2 sin x.

03:08 Now, let us move on to the second part...

Question

a) Find the general solution for the following second-order differential equations: i. d"y/dx" - 9y = 0 ii. 2 d"y/dx" + 16 dy/dx + 32y = 0 iii. d"y/dx" + 4 dy/dx + 5y = 0 b) Solve the following second-order differential equations: d"y/dx" + 10 dy/dx + 21y = 0, y(0) = 2, dy/dx = 2

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