Differential EquationBernoulli Equation Problems 2

Solve the equation $$x^{2} e^{y}\left(\frac{d x}{d y}+x\right)=1$$

$$x^{2} e^{y}\left(\frac{d x}{d y}+x\right)=1 \quad * x^{-2} e^{-y}$$

$$\left(\frac{d x}{d y}+x\right)=x^{-2} e^{-y} \quad * x^{2} $$

$$x^{2}\left(\frac{d x}{d y}+x\right)=e^{-y} \longrightarrow (1)$$

$$x^{2} \frac{d x}{d y}+x^{3}=e^{-y} \cdots (1)$$

let $$z=x^{3} \rightarrow \frac{d z}{d y}=3 x^{2} \frac{d x}{d y} \quad \div 3$$

$$\frac{1}{3} \frac{d z}{d y}=x^{2} \frac{d x}{d y}$$ substitute in $$(1)$$

$$\frac{1}{3} \frac{d z}{d y}+z=e^{-y} \quad * 3$$

$$\frac{d z}{d y}+3 z=3 e^{-y}$$ Linear in $$z$$

$$\mu=e^{\int {3} d y}=e^{3 y}$$

$$\mu {z}=\int \mu \cdot Q(y) d y+c$$

$$e^{3 y} z=\int e^{3 y} \cdot 3 e^{-y} d y+c$$

$$\longrightarrow e^{3 y} x^{3}=\int 3 e^{2 y}+c$$

$$e^{3 y} x^{3}=3 \frac{1}{2} e^{2 y}+c$$

$$\longrightarrow e^{3 y} x^{3}=\frac{3}{2} e^{2 y}+c$$

Solve the equation $$y(x \tan x+\ln y) \mathrm{d} x+\tan x \mathrm{d} y=0$$

$$y(x \tan x+\ln y) d x+\tan x d y=0 \quad \div y$$

$$(x \tan x+\ln y) d x+ \frac{\tan x d y}{y}=0$$

let $$u=\ln y \rightarrow d u=\frac{1}{y} d y$$

$$(x \tan x+u) d x+\tan x d u=0 \quad \div d x$$

$$\tan x \frac{d u}{d x}+u=-x \tan x \quad \div \tan x$$

$$\frac{d u}{d x}+\frac{u}{\tan x}=-x$$

$$\frac{1}{\tan x}=\cot x$$

$$\longrightarrow \frac{d u}{d x}+\cot x u=-x$$

Linear in $$u$$

$$\frac{d u}{d x}+(\cot x) u=-x$$

$$\mu=e^{\int \cot x d x}=e^{\ln |\sin x|}=\sin x$$

$$\mu \cdot u=\int \mu Q(x) d x+C$$

$$\rightarrow \sin (x) \cdot u=\int \sin (x)(-x) d x+c$$

by parts

$$u=-x \quad d u=-1 d x$$
$$v=-\cos x \quad d u=\sin (x)$$

$$\sin x \cdot u=x \cos x-\int-\cos x(-1) d x + c$$

$$\sin x \cdot u=x \cos x-\sin x+c$$