Question

1.       Solve the exact differential equation (𝗑 − 2𝑒^𝑦)𝑑𝑦 + (𝑦 + 𝗑 sin 𝗑)𝑑𝗑 = 0.

2.     Solve the differential equation 𝗑 𝑑𝑦 + 2𝑦 = 𝗑⁴.

Answer 1

1.

$$P(x,y)=y+x\sin x, \dfrac{\partial P}{\partial y}=1$$

$$Q(x,y)=x-2e^y, \dfrac{\partial Q}{\partial x}=1$$

$$\dfrac{\partial P}{\partial y}=1=\dfrac{\partial Q}{\partial x}$$

Write the system of two differential equations that define the function $u(x,y)$

$$\dfrac{\partial u}{\partial x}=y+x\sin x$$

$$\dfrac{\partial u}{\partial y}=x-2e^y$$

$$u=\int(x-2e^y)dy+\varphi(x)$$

$$u=xy-2e^y+\varphi(x)$$

$$\dfrac{\partial u}{\partial x}=y+\varphi'(x)=y+x\sin x$$

$$\varphi'(x)=x\sin x$$

$$\varphi(x)=\int x\sin xdx$$

$$\varphi(x)=-x\cos x+\sin x+C$$

$$u=xy-2e^y-x\cos x+\sin x+C$$

$$-xy+2e^y+x\cos x-\sin x=C$$

2.

$$x\dfrac{dy}{dx}+2y=x^4$$

$$y'+\dfrac{2}{x}y=x^3$$

Integration factor

$$\mu(x)=e^{\int(2/x)dx}=x^2$$

$$x^2y'+2xy=x^5$$

$$d(x^2y)=x^5 dx$$

Integrate

$$\int d(x^2y)=\int x^5 dx$$

$$x^2y=\dfrac{x^6}{6}+C$$

$$y=\dfrac{x^4}{6}+\dfrac{C}{x^2}$$