Question

det. the following d.e. if exact. then dut. the general sol.
a. \\((2x + y - 3)dx+(x - 4y + 1)dy = 0\\)
\\(m\\) \\(n\\)
b. \\((2x + 8y)dx+(8x - 4y)dy = 0\\)
c. \\(2x(1+\frac{1}{y})dx+\frac{y - x^{3}}{y^{2}}dy = 0\\)

Explanation:

Step1: Recall exact - differential - equation condition

For a differential equation \(M(x,y)dx + N(x,y)dy = 0\) to be exact, \(\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}\).

Step2: Analyze part (a)

Given \(M = 2x + y-3\), then \(\frac{\partial M}{\partial y}=1\). Given \(N=x - 4y + 1\), then \(\frac{\partial N}{\partial x}=1\). Since \(\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}\), the DE is exact.
To find the solution, \(\int Mdx=\int(2x + y - 3)dx=x^{2}+xy-3x+h(y)\).
Differentiate with respect to \(y\): \(\frac{\partial}{\partial y}(x^{2}+xy - 3x+h(y))=x+h^{\prime}(y)\).
Since \(\frac{\partial}{\partial y}(x^{2}+xy - 3x+h(y)) = N=x - 4y + 1\), then \(h^{\prime}(y)=-4y + 1\).
Integrating \(h^{\prime}(y)\) with respect to \(y\) gives \(h(y)=-2y^{2}+y+C\).
The general solution is \(x^{2}+xy-3x-2y^{2}+y=C\).

Step3: Analyze part (b)

Given \(M = 2x+8y\), then \(\frac{\partial M}{\partial y}=8\). Given \(N = 8x-4y\), then \(\frac{\partial N}{\partial x}=8\). Since \(\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}\), the DE is exact.
\(\int Mdx=\int(2x + 8y)dx=x^{2}+8xy+h(y)\).
Differentiate with respect to \(y\): \(\frac{\partial}{\partial y}(x^{2}+8xy+h(y))=8x+h^{\prime}(y)\).
Since \(\frac{\partial}{\partial y}(x^{2}+8xy+h(y)) = N = 8x-4y\), then \(h^{\prime}(y)=-4y\).
Integrating \(h^{\prime}(y)\) with respect to \(y\) gives \(h(y)=-2y^{2}+C\).
The general solution is \(x^{2}+8xy-2y^{2}=C\).

Step4: Analyze part (c)

Given \(M = 2x(1+\frac{1}{y})=2x+\frac{2x}{y}\), then \(\frac{\partial M}{\partial y}=-\frac{2x}{y^{2}}\). Given \(N=\frac{y - x^{3}}{y^{2}}=\frac{1}{y}-\frac{x^{3}}{y^{2}}\), then \(\frac{\partial N}{\partial x}=-\frac{3x^{2}}{y^{2}}\). Since \(\frac{\partial M}{\partial y}
eq\frac{\partial N}{\partial x}\), the DE is not exact.

Answer:

a. General solution: \(x^{2}+xy-3x-2y^{2}+y = C\)
b. General solution: \(x^{2}+8xy-2y^{2}=C\)
c. The differential - equation is not exact.