MAT244--2018F > Quiz-2

Q2 TUT 0101, TUT 0501 and TUT 0801

(1/1)

**Victor Ivrii**:

Find an integrating factor and solve the given equation.

$$

1 + \Bigl(\frac{x}{y} - \sin(y)\Bigr)y' = 0.

$$

**Jiexuan Wei**:

Here is my answer to this question.

**Monika Dydynski**:

Find an integrating factor and solve the given equation.

$$1+\left({x \over y}−\sin(y)\right)y′=0.\tag{1}$$

Let $M(x,y)=1$ and $N(x,y)={x \over y}−\sin(y)$.

Then, $M_y={\partial \over \partial y}M(x,y)=0$ and $N_x={\partial \over \partial x}N(x,y)={1 \over y}$.

Notice that $M_y =0\ne{1 \over y} =N_x$, which implies that the given equation is not exact.

We are looking for an integrating factor $\mu(x,y)$ such that after multiplying $(1)$ by $\mu$, the equation becomes exact.

That is, $(\mu M)_y=(\mu N)_x$.

$${\partial \mu \over \partial y }={\partial \over \partial x}\left[\mu\left({x \over y}-\sin(y)\right)\right]$$

$${\partial \mu \over \partial y }={\partial \mu \over \partial x }\left({x \over y}-\sin(y)\right)+\mu {1 \over y}$$

Suppose that $\mu$ is a function of only $y$, we get

$${d \mu \over dy}=\mu {1 \over y}$$

$$\int{1 \over \mu}d \mu=\int{1 \over y}dy $$

$$\ln|\mu|=\ln|y|$$

$$\mu=y$$

Multiplying $(1)$ by the integrating factor $\mu=y$, we get an exact equation,

$$y+(x-y \sin(y))y'=0\tag{2}$$

Since $(2)$ is exact, there exists a solution $\phi (x,y)=C$ such that

$\phi_x(x,y)=y\tag{3}$

$\phi_y(x,y)=x-y \sin(y)\tag{4}$

Integrating $(3)$ with respect to $x$, we get

$$\phi(x,y)=yx+g(y)\tag{5}$$

for some function $g$ of $y$.

Differentiating $(5)$ with respect to $y$, we get

$$\phi_y(x,y)=x+g'(y)\tag{6}$$

Equating $(4)$ with $(6)$, we have

$$x-y\sin(y)=x+g'(y)$$

$$-y\sin(y)=g'(y)$$

$$\int -y\sin(y)dy=\int g'(y)dy$$

$$g(y)=-\int y\sin(y)dy$$

Integration by parts gives,

$$g(y)=y\cos(y)-\sin(y)\tag{7}$$

Substituting $(7)$ into $(5)$,

$$\phi(x,y)=yx+y\cos(y)-\sin(y)$$

Thus, the solutions of the differential equation are given implicitly by

$$yx+y\cos(y)-\sin(y)=C$$

**Yiran Zhu**:

Sorry, I posted to the wrong quiz question

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