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Hello. The question is taken from calculus and the question is find a particular solution to this differential equation. And that differential equation is Y double prime plus four by point plus. For why? That is equal to E. To the minus two weeks divided by X cubed. Okay so in order to so always let us first evaluate the other two solutions from the left hand side. So that will be the square plus 40. I'm just taking that day over the X. Is equal today. So from her day square over the X square is equal to capital D. Square. And Y. Prime is just buy over the X. Okay. And then in the similar way uh Y double prime is a culture data. Why you were the X square? So if we saw this we get the two solution is why uh that is complemented function. So I'm just writing them as Y. One and Y two. Why when is it for two A. To the minus two weeks And why two is equal to X. E. To the minus two weeks. Okay so that is by uh the plus two whole square is equal to zero. So these equal to minus two cuomo minus. So these are two solutions. Now particular integer we can found by evaluating their own cancer, sewing our cash in is violent by one time by two by two prime. So from here let us substitute this value. We get into the minus two weeks. X. E. to the -2 weeks. So And here we get -2 way to the -2 legs. And and the differentiation of the second operator is Second solution is into the -2 weeks minus X two X. E. To the minus two weeks. So if we saw which we get the value of W is equal to E to the minus four X minus two weeks Into the -4 x plus two weeks into the minus four weeks. So that is equal to E to the minus four weeks. Now let us evaluate W one W one is equal to Z O G X G X. S. The function on the right hand side in our case G X. Z to the minus two weeks divided by X cubed. So and this will be why two by 25. So if we substitute this value we get zero into the minus two weeks divided by X. Q. And a little Y two is X. E. To the minus X and Y two minus two rigs. And why to buy Miss E. To the minus two weeks -2 x. Into the -2 weeks. So if we saw which we get it's value is equal to into the minus for this minuses Into the -4 weeks divided by X Square in a similar way that does evaluate the value of W two. W two is equal to Y one by 150 G X. And it was always we get this value is five minutes into the minus two weeks and this will be minus two into the minus two weeks, zero into the minus two weeks over. Excuse So that really gives us value and W two is equal to E. To the minus four X divided by X cubed. Now let us evaluate union one And we want you one is equal to W1 or walk the blue integration with respect to X. So that will be equal to the blue one is E. To the minus four X divided by X. Square. And W. Is equal to Eat with the -4 x. And the X. So that will give us value minus one over X. And U. Two is equal to integration W. Two over W. Into the eggs. So that will be integration. The value of W. Two is into the minus four X. Will be canceled out. And we get one of our xq dx. That will remind us one over X square. Okay minus two minus three plus five minus 1/2 X squared. So particular integral will be Why when you want plus you two x 2 the value of U. One is this. And so that will be E. To the minus two X divided by x Minour to the -2 weeks, divide into X divided by two X squared. So that will be E. To the minus two X divided by X. Just a required value of that particular in table. So hope this clears your doubt and thank
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