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Can this differential equation be solved using separation of variables? (dy)/(dx)=sqrt(7x+y^(2)) Choose 1 answer: (A) Yes (B) No

Can this differential equation be solved using separation of variables?\newlinedydx=7x+y2 \frac{d y}{d x}=\sqrt{7 x+y^{2}} \newlineChoose 11 answer:\newline(A) Yes\newline(B) No

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Full solution

Q. Can this differential equation be solved using separation of variables?\newlinedydx=7x+y2 \frac{d y}{d x}=\sqrt{7 x+y^{2}} \newlineChoose 11 answer:\newline(A) Yes\newline(B) No
  1. Check for Separation of Variables: To determine if the differential equation can be solved using separation of variables, we need to see if we can express the equation in the form of g(y) dy=h(x) dxg(y)\,dy = h(x)\,dx, where g(y)g(y) is a function of yy only and h(x)h(x) is a function of xx only.
  2. Rearrange the Equation: Looking at the given differential equation (dydx=7x+y2)(\frac{dy}{dx} = \sqrt{7x + y^2}), we attempt to separate the variables by moving all yy terms to one side and all xx terms to the other side.
  3. Attempt Variable Separation: We can rewrite the equation as dy7x+y2=dx\frac{dy}{\sqrt{7x + y^2}} = dx. However, we notice that the term 7x+y2\sqrt{7x + y^2} cannot be easily separated into a function of yy times a function of xx because it is a mixed term involving both xx and yy.
  4. Conclusion: Since the term 7x+y2\sqrt{7x + y^2} involves both xx and yy and cannot be separated into a product of a function of xx and a function of yy, the differential equation cannot be solved using the method of separation of variables.

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