Recitation 3

Exact equations

Suppose we have the differential equation $M(x,y)+N(x,y)y'=0$ and a region $D$. If there is a function $\Psi(x,y)$ so that $\Psi_x(x,y) = M(x,y), \Psi_y = N(x,y)$ for all $(x,y)\in D$, then we call the differential equation exact in $D$. In this case, the implicit solution is $\Psi(x,y) = c$ for $(x,y)\in D$, and we call $\Psi(x,y)$ the potential function.

Theorem (Test for exact differential equation) Suppose the region $D$ is simply connected. The differential equation $M(x,y)+N(x,y)y'=0$ is exact in $D$ if and only if $M_y(x,y) = N_x(x,y)$ at each $(x,y)\in D$.

Problem 1 Determine whether each of the equations is exact. If it is exact, find the solution. (a) $3x^2-2xy+2+(6y^2-x^2+3)y'=0$; (b) $(e^x\sin y-2y\sin x)+(e^x\cos y+2\cos x)y'=0$.

Solution 1(a) Since $\frac{\partial 3x^2-2xy+2}{\partial y} = -2x$ and $\frac{\partial 6y^2-x^2+3}{\partial x} = -2x$ agree for all $(x,y)$, the DE is exact. Let $\Psi(x,y)$ be the potential function. Since $\Psi_x = 3x^2 - 2xy + 2$, $\Psi = x^3 - x^2y + 2x + C(y)$. Since $\Psi_y = -x^2 + C'(y) = 6y^2 - x^2 + 3$, $C(y) = 2y^3+3y$. The implicit solution to the DE is then $x^3-x^2y+2x+2y^3+3y = C$.

Solution 1(b) Since $\frac{\partial e^x\sin y - 2y\sin x}{\partial y} = e^x\cos y - 2\sin x$ and $\frac{\partial e^x\cos y + 2\cos x}{\partial x} = e^x\cos y - 2\sin x$ agree for all $(x,y)$, the DE is exact. Let $\Psi(x,y)$ be the potential function. Since $\Psi_x = e^x\sin y - 2y\sin x$, $\Psi = e^x\sin y + 2y\cos x + C(y)$. Since $\Psi_y = e^x\cos y + 2\cos x + C'(y) = e^x\cos y + 2\cos x$, $C(y) = 0$. The implicit solution to the DE is then $e^x\sin y + 2y\cos x = C$.

Idea It is sometimes possible to convert a differential equation that is not exact into an exact equation by multiplying the equation by a suitable integrating factor.

Problem 2 Given that the integrating factor depends on $y$ only, solve $y+(2x-ye^y)y'=0$.

Solution Let $\mu(y)$ be the integrating factor such that $\mu(y)y+\mu(y)(2x-ye^y)y'=0$ becomes exact. By the test for exactness, $(\mu(y)y)_y = (\mu(y)(2x-ye^y))_x$, equivalently, $\mu'(y)y + \mu(y) = 2\mu(y)$. Thus $\mu(y)$ satisfies $\mu' = y\mu(y)$ and so $\mu(y) = y$. Therefore $y^2 + (2xy - y^2e^y)y'=0$ is exact. Let $\Psi$ be the potential function. Since $\Psi_x = y^2$, $\Psi = xy^2 + C(y)$. Since $\Psi_y = 2xy + C'(y) = 2xy - y^2e^y$, $C(y) = \int -y^2e^y dy = -e^y(y^2-2y+2)$. The solution to the DE is then $xy^2 - e^y(y^2-2y+2) = C$.

Remark In general, $M$ and $N$ have to satisfy certain condition so that $M(x,y)+N(x,y)y'=0$ has an integrating factor $\mu$ that depends on $y$ only. Let $\mu(y)$ be an integrating factor. This means $\mu(y)M(x,y) + \mu(y)N(x,y)y' = 0$ is exact. Thus $(\mu(y)M(x,y))_y = (\mu(y)N(x,y))_x$. The left hand side equals $\mu'M + \mu M_y$, and the right hand side equals $\mu N_x$. This shows $\mu' M + \mu M_y = \mu N_x$, that is $\mu' = \frac{N_x - M_y}{M} \mu$. The condition for the existence of such $\mu$ is that $\frac{N_x - M_y}{M}$ depends only on $y$.

Orthogonal families of curves

Definition Consider two families of curves $\mathcal{F}$ and $\mathcal{G}$. We say that they are orthogonal whenever any curve from $\mathcal{F}$ intersects any curve from $\mathcal{G}$, and the two curves are orthogonal at the point of intersection.

Example The families $y = kx$ and $x^2+y^2=r^2$ are orthogonal.

Question Given a family of curves $\mathcal{F}$, is it possible to find a family of curves which is orthogonal to $\mathcal{F}$?

Example Each line through the origin, that is, $y = ax$, we have $y' = a$, and so $xy' = ax = y$. In other words, every line in the given family satisfies the DE $xy' = y$. Rewrite the DE as $y' = y/x$. The slope diagram is as follows:

Imagine that each little segment in the slope diagram turns 90 degrees:

On one hand, the 2nd slope diagram satisfies the DE $y' = -1/(y/x) = -x/y$ (why?). On the other hand, a solution (an integral curve) to the 2nd slope will intersect every line through the origin at a right angle (again, why?). This means the family of curves that is orthogonal to the family of lines through the origin satisfies the DE $y' = -x/y$. By separation of variables, $x + yy' = 0$, and so $x^2+y^2 = C$. This represents the family of circles centered at the origin.

We summarize the above discussion as follows.

Step 1 Find the differential equation associated to the given family of curves $\mathcal{F}$ and rewrite this differential equation in the explicit form $\frac{dy}{dx} = f(x,y)$.

Step 2 Write down the differential equation $\frac{dy}{dx} = \frac{-1}{f(x,y)}$ associated to the orthogonal family $\mathcal{G}$, and solve the differential equation.

Optional step Give a geometric view of the two families.

Exercise 1 Find the orthogonal family of the family of circles $x^2+y^2=2cx$.

Exercise 2 (Hard) Find the orthogonal family of the family of cardiods $$x^2+y^2+ax=a\sqrt{x^2+y^2}.$$