Question

In Exercises 1–22, solve the differential equation.x dy - (x⁴ - y) dx = 0

Accepted Answer

Rewrite the given differential equation in the form $$M(x,y)\,dx + N(x,y)\,dy = 0. $$Here, the equation is $$x\,dy - (x$$^{4}$$ - y)\,dx = 0$$, which can be rearranged as $$-(x^{4} - y$$)\,dx + x\,dy = 0$$. So, M(x$$,y) = $$-(x^{4} - y$$)$$ and N(x$$,y) = x$$. Check if the differential equation is exact by verifying if $$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$$. Compute $$\frac{\partial M}{\partial y}$$ and $$\frac{\partial N}{\partial x}$$. If the equation is not exact, find an integrating factor that depends on either x$ or y$ to make it exact. To do this, use the formula for integrating factors: for example, if $$\frac{\partial}{\partial y}(M)/N$$ depends only on x$, then the integrating factor is a function of x$. Once the equation is exact (either originally or after multiplying by the integrating factor), find the potential function $$\Psi(x,y)$$ such that $$\frac{\partial \Psi}{\partial x} = M$$ and $$\frac{\partial \Psi}{\partial y} = N$$. Integrate M$ with respect to x$ and include an unknown function of y$, then differentiate this result with respect to y$ and set equal to N$ to solve for the unknown function. Write the implicit solution as $$\Psi(x,y) = C$$, where C$ is a constant. This represents the general solution to the differential equation.

In Exercises 1–22, solve the differential equation.

x dy - (x⁴ - y) dx = 0

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Key Concepts

Differential Equations

Exact Differential Equations

Integrating Factor