Embibe Experts Solutions for Chapter: Differential Equations, Exercise 4: EXERCISE-4

State the order and degree of the following differential equation:

${\left[\frac{d^{2}x}{ d{t}^2}\right]}^3+{\left[\frac{dx}{dt}\right]}^4-xt=0$

State the order and degree of the following differential equation:

$\frac{d^{2}y}{d{x}^2}={\left[1+{\left(\frac{dy}{dx}\right)}^2\right]}^{3/2}$

Solve:

$\frac{dy}{ dx}=\frac{x^2+xy}{x^2+y^2}$

Solve:

$\left(x^3-3x{y}^2\right)dx=\left(y^3-3x^{2}y\right)dy$

Given two curves $y=f\left(x\right)$, where $f\left(x\right)>0$, passing through the points $(0,1) \& y={\int }_{-\infty }^{x}f\left(t\right)dt$ passing through the points $(0,\frac{1}{2})$. The tangents drawn to both curves at the points with equal abscissas intersect on the $x$-axis. Find the curve $f\left(x\right)$.

Find the orthogonal trajectory for $y=a{x}^2$  where '$a$' is the parameter.

Find the orthogonal trajectory for $\cos y=a{e}^{-x}$  where '$a$' is the parameter.

Let $f\left(x,y,c_1\right)=0$ and $f\left(x,y,c_2\right)=0$ define two integral curves of a homogeneous first order differential equation. If $P_1$ and $P_2$ are respectively the points of intersection of these curves with an arbitrary line, $y=mx$ then prove that the slopes of these two curves at $P_1$ and $P_2$ are equal.