Solve the following differential equation. (1+y*2)dx+(x-e*(-tan*(-1)y)dy?

Answer 1

=> dx/dy + x/(1+y^2) = e^(-tan^(-1)y)/(1+y^2)
So,
it is of form dx/dy +Px = Q
Integrating factor (IF)= e^(integral(P)dy) = e^(tan^(-1)y)
Sol is
=> x*IF = Integral(Q*IF)dy

=> x*e^(tan^(-1)y) = integral(1/(1+y^2)) = tan^(-1)y +C
=> x = tan^(-1)y * e^(-tan^(-1)y) + C * e^(-tan^(-1)y)

Answer 2

some answerers have wondered the certain answer with the certain critical, those words have diverse meanings and are actual not interchangeable. Rewrite this differential equation in favourite kind: (D² + a million)y = ?? + x y'' + y = ?? + x locate the complementary function with the help of fixing the auxliary equation: y'' + y = 0 m² + a million = 0 m² = -a million m = ±i y? = C?sinx + C?cosx locate the certain critical with the help of comparing coefficients: y? = Ax + B + C?? y?' = A + C?? y?'' = C?? y?'' + y? = ?? + x C?? + Ax + B + C?? = ?? + x Ax + B + 2C?? = x + ?? A = a million B = 0 2C = a million C = ½ y? = x + ?? / 2 locate the overall answer with the help of combining those 2 elements: y = y? + y? y = C?sinx + C?cosx + x + ?? / 2 y = x + ?? / 2 + C?sinx + C?cosx

Answer 3

We need the rest of the dy part!

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Answer 4

=> dx/dy + x/(1+y^2) = e^(-tan^(-1)y)/(1+y^2)
So,
its integrating factor (IF)= e^(int(P)dy) = e^(tan^(-1)y)
Sol is
=> x*IF = int (Q*IF)dy
=> x*e^(tan^(-1)y) = int (1/(1+y^2)) = tan^(-1)y +C
=> x = tan^(-1)y * e^(-tan^(-1)y) + C * e^(-tan^(-1)y)

here, "int" is a integral with respect to x

Answer 5

please post the question correctly