First-Order Differential Equations Flashcards

Question 1

What is a first-order ordinary differential equation (ODE)?

Accepted Answer

An equation relating an unknown function $y(x)$ and its first derivative $y'$, of the form $F(x,y,y')=0$ (or solved as $y'=f(x,y)$).

Question 2

State the integrating factor formula for $y' + P(x)y = Q(x)$.

Accepted Answer

$$\mu(x)=e^{\int P(x)\,dx}.$$ Multiplying the ODE by $\mu$ makes the left side $(\mu y)'$.

Question 3

How do you test exactness and solve $M(x,y)\,dx + N(x,y)\,dy=0$ when it is exact?

Accepted Answer

Test: $$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}.$$ If exact, find $\psi$ with $\psi_x=M$ by integrating in $x$: $\psi=\int M\,dx + g(y)$, then use $\psi_y=N$ to determine $g(y)$. The solution is $\psi(x,y)=C$.

First-Order Differential Equations Flashcards

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