The order of the differential equation $\frac{d^4 y}{d x^4}-\sin \left(\frac{d^2 y}{d x^2}\right)=5$

The sum of the order and the degree of the differential equation $$\frac{d}{d x}\left(\left(\frac{d y}{d x}\right)^3\right)$$ is:

(a) Find the particular solution of the differential equation $\frac{d y}{d x}-2 x y=3^{x^2} e^{x^2} ; y(0)=5$.

OR

(b) Solve the following differential equation: $$ x^2 d y+y(x+y) d x=0 $$

Case Study III

An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form $$\frac{d y}{d x}=F(x, y)$$ is said to be homogeneous if $$\mathrm{F}(x, y)$$ is a homogeneous function of degree zero, whereas a function $$\mathrm{F}(x, y)$$ is a homogenous function of degree $$n$$ if $$\mathrm{F}(\lambda x, \lambda y)=\lambda^n F(x, y)$$. To solve a homogeneous differential equation of the type $$\frac{d y}{d x}=\mathrm{F}(x, y)= g\left(\frac{y}{x}\right)$$, we make the substitution $$y=v x$$ and then separate the variables.

Based on the above, answer the following questions:

(i) Show that $$\left(x^2-y^2\right) d x+2 x y d x=0$$ is a differential equation of the type $$\frac{d y}{d x}=g\left(\frac{y}{x}\right)$$.

(ii) Solve the above equation to find its general solution.