Example 3. Solve the equation $2 x y \ln y d x+\left(x^{2}+y^{2} \sqrt{y^{2}+1}\right) d y=0$.
Solution. We have $P=2 x y \ln y, \quad Q=x^{2}+y^{2} \sqrt{y^{2}+1}, \quad \frac{\partial P}{\partial y}=$ $=2 x \ln y+2 x, \frac{\partial Q}{\partial x}=2 x, \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}=-2 x \ln y, \frac{1}{P}\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)=-\frac{1}{y}$. We find the integrating factor: $\ln \mu(y)=-\int \frac{d y}{y}=-\ln y$. Hence, $\mu(y)=\frac{1}{y}$. After multiplying the equation term by term by $\mu(y)=\frac{1}{y}$, we obtain the equation in exact differentials $2 x \ln y d x+\left(\frac{x^{2}}{y}+y \sqrt{y^{2}+1}\right) d y=0$. Its general solution is $x^{2} \ln y+\frac{1}{3}\left(y^{2}+1\right)^{3 / 2}=C_{1}$.

Answer. $3 x^{2} \ln y+\sqrt{\left(y^{2}+1\right)^{3}}=C$.

Note. Some equations admit integrating factors depending on both variables $x$ and $y$. To find such factors, the equation must be multiplied term by term by $\mu(x, y)$ and the possibilities analyzed for when the new equation can be an equation in exact differentials. We do not consider such equations.

## Exercises

Solve the equations in exact differentials.

1. $3 x^{2} e^{y} d x+\left(x^{3} e^{y}-1\right) d y=0$.
2. $x\left(2 x^{2}+y^{2}\right)+y\left(x^{2}+2 y^{2}\right) y^{\prime}=0$.
3. $\left(3 x^{2}+6 x y^{2}\right) d x+\left(6 x^{2} y+4 y^{3}\right) d y=0$.
4. $(x \cos 2 y+1) d x=x^{2} \sin 2 y d y$.
5. $\left(x^{3}-3 x y^{2}+2\right) d x-\left(3 x^{2} y-y^{2}\right) d y=0$.
6. $\left(\frac{\sin 2 x}{y}+x\right) d x+\left(y-\frac{\sin ^{2} x}{y^{2}}\right) d y=0$.
7. $\frac{x d x+y d y}{\sqrt{x^{2}+y^{2}}}+\frac{x d y-y d x}{x^{2}}=0$.

Solve the equations by finding a suitable integrating factor.
8. $\left(2 x y+x^{2} y+\frac{y^{4}}{3}\right) d x+\left(x^{2}+\frac{4}{3} y^{3}\right) d y=0$.
9. $\left(2 x^{2} y+2 y+5\right) d x+\left(2 x^{3}+2 x\right) d y=0$.
10. $\left(2 x y^{2}-3 y^{3}\right) d x+\left(7-3 x y^{2}\right) d y=0$.

## Answers

1. $x^{3} e^{y}-y=$ C. 2. $x^{4}+x^{2} y^{2}+y^{4}=$ C. 3. $x^{3}+3 x^{2} y^{2}+y^{4}=$ =C. 4. $x^{2} \cos 2 y+2 x=C . \quad$ 5. $\frac{x^{4}}{4}-\frac{3}{2} x^{2} y^{2}+2 x+\frac{y^{3}}{3}=C$. 6. $\frac{\sin ^{2} x}{y}+\frac{x^{2}+y^{2}}{2}=C$. 7. $\sqrt{x^{2}+y^{2}}+\frac{y}{x}=$ C. 8. $y e^{x}\left(x^{2}+\frac{y^{3}}{3}\right)=C$.

![](https://cdn.mathpix.com/cropped/2024_05_22_446938f1e4808e7d6e07g-087.jpg?height=82&width=1114&top_left_y=416&top_left_x=83)
10. $x^{2}-\frac{7}{y}-3 x y=$ C. Hint. $\mu(y)=\frac{1}{y^{2}}$.

## § 6. First-Order Differential Equations Not Solved for the Derivative