Allow y(x) be the solution for the following initial value problem and then find y(e):  dy/dx = (lnx)^6/x*y^2 ,  y(1)= 3

Q: If ( 1 + x²)(1 + y²) - px² = 0, where p is a constant, show that xy ( 1 + x²)dy/dx = ( 1 + y²).

A: Given: To Find:

Q: Solve of the following Clairaut equation y=((x+4)y')+(y')^2

A:

Q: Find the equation of the function f whoseg 3 f'(x) = (0, 4) %3D -- 1 + e

A:

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.