Question

Use linear substitution to solve the following first-order differential equation

𝑑𝑦/𝑑𝑥=(2𝑥+𝑦)/(2𝑥+𝑦+1)

Answer 1

$$u=2x+y$$

$$\dfrac{du}{dx}=2+\dfrac{dy}{dx}$$

Substitute

$$\dfrac{du}{dx}-2=\dfrac{u}{u+1}$$

$$\dfrac{du}{dx}=\dfrac{3u+2}{u+1}$$

$$\dfrac{u+1}{3u+2}du=dx$$

Integrate

$$\int\dfrac{u+1}{3u+2}du=\int dx$$

$$\int\dfrac{u+1}{3u+2}du=\dfrac{1}{3}\int\dfrac{3u+2}{3u+2}du+\dfrac{1}{3}\int\dfrac{1}{3u+2}du$$

$$=\dfrac{1}{3}u+\dfrac{1}{9}\ln|(3u+2)|+C_1$$

$$\dfrac{1}{3}u+\dfrac{1}{9}\ln|(3u+2)|=x+\dfrac{1}{9}\ln C$$

$$3u+\ln|(3u+2)|=9x+\ln C$$

$$(3u+2)e^{3u}=Ce^{9x}$$

Then

$$(6x+3y+2)e^{6x+2y}=Ce^{9x}$$

$$(6x+3y+2)e^{2y}=Ce^{3x}$$