Find two linearly independent solutions of y 4xy 0 of the...

Name: find two linearly independent solutions of y 4xy 0 of the form y1 1 a3x3 6x2 y2 x b1x2 b7ac7 enter the first few coefficients a3 a6 b4 by p 0666 01776 0333 00476 97722
Uploaded: 2023-07-18T03:23:08-08:00
Duration: 6 min 15 s
Channel: Madhur L
Description: find two linearly independent solutions of y 4xy 0 of the form y1 1 a3x3 6x2 y2 x b1x2 b7ac7 enter the first few coefficients a3 a6 b4 by p 0666 01776 0333 00476 97722

Find two linearly independent solutions of y" + 4xy = 0 of the form y₁ = 1+ a3x³ + α6x² + ... Y₂ = x + b₁x² +b7ac7 +... Enter the first few coefficients: a3 |||| a6 = b4 ||_|| by= P -0.666 0.1776 -0.333 0.0476

Transcript

00:01 Hello everyone, in this problem we are given with the differential equation y' ' plus 4xy to be equal to 0 and it is given that it has two linearly independent solutions.

00:13 The first one as y1 to be equal to 1 plus a3x3 plus a6x6 plus and so on and another be y2.

00:26 So now let us first solve for y1 by finding the values of a3 and a6 and with the value of y2 we can solve for b4 and b7.

00:40 So now differentiating this equation y1 to be y1 ' so we get this value to be 3a3x square plus 6a6x power 5 plus and so on.

01:02 Now again differentiating we have y1' ' to be equal to 6a3x plus 30a6x power 4 plus and so on.

01:15 Now substituting the value of y1' ' and y ' in equation 1 so we have the equation 1 will become 6a3x plus 30a6x power 4 plus and so on plus 4x y1 ' sorry y1 to be equal to 1 plus a3x3 plus a6x power 6 plus and so on to be equal to 0.

01:55 So now simplifying this we have this value to be 6a3 plus 4 of x plus 30a6 plus 4a3 of x power 4 and so on to be equal to 0.

02:10 So now equating the coefficients of x and x power 4 on both sides so we have 6a3 plus 4 to be equal to 0 so from this we get the value of a3 to be equal to minus 2 by 3 and 3a6 plus 4a3 to be equal to 0 so from this we get 3a6 to be equal to minus 4a3.

02:46 Now substituting the value of a3 to be minus 4 multiplied by minus 2 by 3 so we get this value to be 8 by 3 so from this we get the value of a6 to be 8 by 9.

03:01 So here these are the required values of a3 and a6.

03:09 Now let us have this another solution y2 to be x plus b1 x power 4 sorry this b4 x power 4 plus b7 x power 7 plus and so on is linearly independent...

Question

Find two linearly independent solutions of y" + 4xy = 0 of the form y₁ = 1+ a3x³ + α6x² + ... Y₂ = x + b₁x² +b7ac7 +... Enter the first few coefficients: a3 |||| a6 = b4 ||_|| by= P -0.666 0.1776 -0.333 0.0476

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