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Hello everyone.
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From the question we are given that the differential equation 2y dx plus x into log x minus log y minus 2, dy is equal to 0 with initial value y 1 is equal to e.
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Or we can write it as 2 into dx upon dy plus x upon y into log x upon y minus 2 is equal to 0.
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Let x equals to v into y.
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Differential both side with respect to y then we get dx upon dy is equal.
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To v plus y into dv upon d y.
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Now put the value of x and d x upon d y in this equation.
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Then we have 2 into v plus y into dv upon d y plus v 0.
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This implies 2v plus 2y dv upon dy plus 2y plus v log v minus 2v is equal to 0.
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By simplifying this we get 2y dv upon dy plus v into log v equals to 0.
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By separation of variables we can write it as 2dv upon v 0 .m.
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V.
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0.
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D .m.
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Is equal to minus d .y upon y.
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Now integrating both sides to solve this integration, let log v is equal to z...