CE Board Exam Randomizer

⬅ Back to Differential Equations
📘 Key Concepts: Exact Equations

The equation $M(x,y)\,dx + N(x,y)\,dy = 0$ is exact if:

$$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$$

If exact, there exists $F(x,y)$ such that $dF = M\,dx + N\,dy$, and the solution is $F(x,y) = C$.

Finding F: Integrate $M$ with respect to $x$ (treating $y$ as constant), then differentiate with respect to $y$ and match to $N$ to find any $y$-only function $g(y)$.

If not exact: Find an integrating factor $\mu$ such that multiplying through makes it exact.
— If $\dfrac{\partial M/\partial y - \partial N/\partial x}{N}$ is a function of $x$ only: $\mu = e^{\int (\cdot)\,dx}$
— If $\dfrac{\partial N/\partial x - \partial M/\partial y}{M}$ is a function of $y$ only: $\mu = e^{\int (\cdot)\,dy}$

Problem 1: Basic Exact Equation

Test for exactness and solve: $(2xy + 3)\,dx + (x^2 - 4y)\,dy = 0$.

DE – Exact – Problem 1 – Setup DE – Exact – Problem 1 – Diagram DE – Exact – Problem 1 – Diagram
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Problem 2: Exact Equation with Initial Condition

Solve $(2xy^2 + y)\,dx + (2x^2y + x)\,dy = 0$, $y(1) = 2$.

DE – Exact – Problem 2 – Setup DE – Exact – Problem 2 – Diagram DE – Exact – Problem 2 – Diagram
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Problem 3: Exact Equation with Exponential Terms

Test for exactness and solve: $(e^x \sin y + 2x)\,dx + (e^x \cos y)\,dy = 0$.

DE – Exact – Problem 3 – Setup DE – Exact – Problem 3 – Diagram DE – Exact – Problem 3 – Diagram
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Problem 4: Non-Exact — Find Integrating Factor

Show that $(3xy + y^2)\,dx + (x^2 + xy)\,dy = 0$ is not exact. Find an integrating factor $\mu(x)$ and solve the resulting exact equation.

DE – Exact – Problem 4 – Setup DE – Exact – Problem 4 – Diagram DE – Exact – Problem 4 – Diagram
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