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67_pdfsam_math 54 differential equation solutions odd

# 67_pdfsam_math 54 differential equation solutions odd -...

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Exercises 2.4 15. This differential equation is expressed in the variables r and θ . Since the variables x and y are dummy variables, this equation is solved in exactly the same way as an equation in x and y . We will look for a solution with independent variable θ and dependent variable r . We see that the differential equation is expressed in the differential form M ( r, θ ) dr + N ( r, θ ) = 0 , where M ( r, θ ) = cos θ and N ( r, θ ) = r sin θ + e θ . This implies that M θ ( r, θ ) = sin θ = N r ( r, θ ) , and so the equation is exact. Therefore, to solve the equation we need to find a function F ( r, θ ) that has cos θ dr + ( r sin θ + e θ ) as its total differential. Integrating M ( r, θ ) with respect to r we see that F ( r, θ ) = cos θ dr = r cos θ + g ( θ ) F θ ( r, θ ) = r sin θ + g ( θ ) = N ( r, θ ) = r sin θ + e θ . Thus we have that g (
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