Question

In Exercises 23–28, solve the initial value problem.x dy + (y - cos x) dx = 0, y(π/2) = 0

Accepted Answer

Rewrite the given differential equation in the form $$ M(x,y) \, dx + N(x,y) \, dy = 0 . $$Here, the equation is $$ x \, dy + (y - \cos x) \, dx = 0 $$, so rearranged as $$ (y - \cos x) \, dx + x \, dy = 0 . $$Check if the differential equation is exact by computing the partial derivatives $$ \frac{\partial M}{\partial y} $$ and $$ \frac{\partial N}{\partial x} $$, where $$ M = y - \cos x $$ and $$ N = x . If $$they are equal, the equation is exact. If the equation is exact, find the potential function $$ \Psi(x,y) $$ such that $$ \frac{\partial \Psi}{\partial x} = M = y - \cos x $$ and $$ \frac{\partial \Psi}{\partial y} = N = x . $$Integrate $$ M $$ with respect to $$ x  to $$find $$ \Psi(x,y)  up to a $$function of $$ y . $$Differentiate the expression for $$ \Psi(x,y) $$ with respect to $$ y $$ and set it equal to $$ N = x  to $$determine the function of $$ y $$ introduced during integration. Use the initial condition $$ y(\pi/2) = 0  to $$find the constant of integration in the implicit solution $$ \Psi(x,y) = C $$, and write the final implicit solution to the initial value problem.

In Exercises 23–28, solve the initial value problem.
x dy + (y - cos x) dx = 0, y(π/2) = 0

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Key Concepts

Initial Value Problem (IVP)

Exact Differential Equations

Integrating Factor