The degree of the differential equation $\sqrt{1+\left(\frac{d y}{d x}\right)^2}=x$ is ................. .
Given differential equation is $\sqrt{1+\left(\frac{d y}{d x}\right)^2}=x$
So, degree of this equation is two.
The number of arbitrary constants in the general solution of a differential equation of order three is .......... .
There are three arbitrary constants.
$\frac{d y}{d x}+\frac{y}{x \log x}=\frac{1}{x}$ is an equation of the type ........... .
Given differential equation is $\frac{d y}{d x}+\frac{y}{x \log x}=\frac{1}{x}$
The equation is of the type $\frac{d y}{d x}+P y=Q$
General solution of the differential equation of the type is given by ............ .
Given differential equation is
$$\frac{d x}{d y}+P_1 x=Q_1$$
The general solution is
$$x \cdot \mathrm{IF}=\int Q(\mathrm{IF}) d y+C \text { i.e., } x \mathrm{e}^{\int P d y}=\int Q\left\{e^{\int P d y}\right\} d y+C$$
The solution of the differential equation $\frac{x d y}{d x}+2 y=x^2$ is ............. .
Given differential equation is
$$x \frac{d y}{d x}+2 y=x^2 \Rightarrow \frac{d y}{d x}+\frac{2 y}{x}=x$$
This equation of the form $\frac{d y}{d x}+P y=Q$.
$$\therefore\quad \mathrm{IF}=e^{\int \frac{2}{x} d x}=e^{2 \log x}=x^2$$
$$\begin{aligned} &\begin{array}{ll} \text { The general solution is }\\ y x^2=\int x \cdot x^2 d x+C \\ \Rightarrow \quad y x^2=\frac{x^4}{4}+C \\ \Rightarrow \quad y=\frac{x^2}{4}+C x^{-2} \end{array} \end{aligned}$$