HardMathematicsHigh School (11–12)Functions and CalculusNumerical

A mathematics club studies the family of curves given by the equation (x² + y² + x) dx + xy dy = 0…

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A mathematics club studies the family of curves given by the equation $(x^2 + y^2 + x)\,dx + xy\,dy = 0$ for $x > 0$ . The equation is not exact as it stands, but one factor that depends on $x$ alone makes it exact. A member finds the solution curves in the form $\dfrac{x^4}{4} + \dfrac{x^3}{3} + \dfrac{x^2y^2}{2} = C$ and picks the curve through $(1, 1)$ . Find the integrating factor at $x = 4$ , the constant $C$ to four decimal places, and the positive value of $y$ on the curve at $x = 0.5$ , to four decimal places.

Need a reminder? The key ideaShow

$M\,dx + N\,dy = 0$ is exact when $\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}$ , and then $F(x, y) = C$ with $F_x = M$ and $F_y = N$ . If $\dfrac{M_y - N_x}{N}$ depends on $x$ only, $\mu = e^{\int \frac{M_y - N_x}{N}dx}$ . If $\dfrac{N_x - M_y}{M}$ depends on $y$ only, $\mu = e^{\int \frac{N_x - M_y}{M}dy}$ . For homogeneous $M$ , $N$ use $\mu = \dfrac{1}{Mx + Ny}$ .

The worked solution

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