Question

Solve each of the following by two methods (see Exercise \(21(b))\) : (a) \(\left(x^{2}+2 y^{2}\right) d x+\left(4 x y-y^{2}\right) d y=0\) (b) \(\left(2 x^{2}+2 x y+y^{2}\right) d x+\left(x^{2}+2 x y\right) d y=0\)

Short answer

For equation (a): \[\left(x^{2}+2 y^{2}\right) d x+\left(4 x y-y^{2}\right) d y=0\] The solution is: \[x^2 + 2y^2 + 2x(y^2) - \frac{y^3}{3} = C\] For equation (b): \[\left(2 x^{2}+2 x y+y^{2}\right) d x+\left(x^{2}+2 x y\right) d y=0\] The solution is: \[2x^2 + 2xy + y^2 + x^2y + x(y^2) = C\]

Step-by-step solution

Test for exactness

Check if \(M = x^2 + 2y^2\) and \(N = 4xy - y^2\) are exact. Compute the partial derivatives \(\frac{\partial M}{\partial y} = 4y\) \(\frac{\partial N}{\partial x} = 4y\) Since \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\), the given equation is an exact differential equation.

Exact differential equation method

Rewrite the equation as: \(d(x^2 + 2y^2) + (4xy - y^2) dy = 0\) Integrate both sides with respect to x: \(\int [d(x^2 + 2y^2) + (4xy - y^2) dy] \,dx\) \(x^2 + 2y^2 + \int (4xy - y^2) dy = C\) Now, integrate \(4xy - y^2\) with respect to y: \(x^2 + 2y^2 + 2x(y^2) - \frac{y^3}{3} = C\)

Integrating factor method

Since the equation is exact, there is no need to use the integrating factor method. The solution is: \(x^2 + 2y^2 + 2x(y^2) - \frac{y^3}{3} = C\) #### Part (b) ####

Test for exactness

Check if \(M = 2x^2 + 2xy + y^2\) and \(N = x^2 + 2xy\) are exact. Compute the partial derivatives \(\frac{\partial M}{\partial y} = 2x + 2y\) \(\frac{\partial N}{\partial x} = 2x + 2y\) Since \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\), the given equation is an exact differential equation.

Exact differential equation method

Rewrite the equation as: \(d(2x^2 + 2xy + y^2) + (x^2 + 2xy) dy = 0\) Integrate both sides with respect to x: \(\int [d(2x^2 + 2xy + y^2) + (x^2 + 2xy) dy] \,dx\) \(2x^2 + 2xy + y^2 + \int (x^2 + 2xy) dy = C\) Now, integrate \(x^2 + 2xy\) with respect to y: \(2x^2 + 2xy + y^2 + x^2y + x(y^2) = C\)

Integrating factor method

Since the equation is exact, there is no need to use the integrating factor method. The solution is: \(2x^2 + 2xy + y^2 + x^2y + x(y^2) = C\)