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Show that an implicit solution of 2x sin²(y) dx - (x² + 18) cos(y) dy = 0 is given by In(x² + 18) + csc(y) = C. Differentiating In(x² + 18) + csc(y) = C we get 2x x² +18 -csc(y) cot(y) ) dx = 0 or 2x sin²(y) dx + 2x sin²y dx − (x² + 18) cos y dy = 0. X Find the constant solutions, if any, that were lost in the solution of the differential equation. (Let k represent an arbitrary integer.) y = Ka+ 7/7/2

Show that an implicit solution of 2x sin²(y) dx - (x² + 18) cos(y) dy = 0 is given by In(x² + 18) + csc(y) = C. Differentiating In(x² + 18) + csc(y) = C we get 2x x² +18 -csc(y) cot(y) ) dx = 0 or 2x sin²(y) dx + 2x sin²y dx − (x² + 18) cos y dy = 0. X Find the constant solutions, if any, that were lost in the solution of the differential equation. (Let k represent an arbitrary integer.) y = Ka+ 7/7/2

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Show that an implicit solution of 2x sin²(y) dx - (x² + 18) cos(y) dy = 0 is given by In(x2 + 18) + csc(y) = C. Differentiating In(x² + 18) + csc(y) = C we get x² 2x +18 -csc (y) cot(y) -= 0 or 2x sin²(y) dx + (2x sin²y dx − (x² + 18) cos y dx X Find the constant solutions, if any, that were lost in the solution of the differential equation. (Let k represent an arbitrary integer.) π y = K₁ + 7/2/2 )dy = 0.
Transcribed Image Text:Show that an implicit solution of 2x sin²(y) dx - (x² + 18) cos(y) dy = 0 is given by In(x2 + 18) + csc(y) = C. Differentiating In(x² + 18) + csc(y) = C we get x² 2x +18 -csc (y) cot(y) -= 0 or 2x sin²(y) dx + (2x sin²y dx − (x² + 18) cos y dx X Find the constant solutions, if any, that were lost in the solution of the differential equation. (Let k represent an arbitrary integer.) π y = K₁ + 7/2/2 )dy = 0.
Expert Solution
Step 1

Given that 

Equation : 2x sin2(y)dx - (x2+18) cos(y) dy = 0

and Implicit solution : ln(x2+18) +csc(y) = c

Now we have to find the constant solution .

consider , 

              ln(x2+18) +csc(y) = c

now, differencing  we get 

      2x /(x2+18) + ( -csc(y) . cot(y) ) dy/dx = 0

or, 2x /(x2+18) - ( 1/ sin(y) cos(y)/sin(y) ) dy/dx = 0

or, 2x. sin2(y) - (x2+18) cos(y) dy/dx = 0 ..................(i)

 

 

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