How do you find the differential equation of the family of curves?

Determine the differential equation for the family of curves defined by the equation y=ex+C. Solution. Differentiating the given equation with respect to x gives: y′=ex+C.

Can every family of curves be given by an ode?

Every one-parameter family of curves is the general solution of some first order ordinary differential equation. Conversely, every first order ordinary differential equation has as its general solution some one-parameter family of curves.

How do you find orthogonal trajectory for a family of curves?

The algorithm includes the following steps:

  1. Construct the differential equation G(x,y,y′)=0 for the given family of curves g(x,y)=C.
  2. Replace y′ with (−1y′) in this differential equation.
  3. Solve the new differential equation to determine the algebraic equation of the family of orthogonal trajectories f(x,y)=C.

What do you mean by family of curves?

A family of curves is a set of curves, each of which is given by a function or parametrization in which one or more of the parameters is variable. In general, the parameter(s) influence the shape of the curve in a way that is more complicated than a simple linear transformation.

What is the order of differential equation representing family of curves and being arbitrary constants?

The family of curves whose differential equation has to be obtained is represented by $v = \dfrac{A}{r} + B$ is given where A and B are arbitrary constants. We have eliminated both the arbitrary constants A and B.

What is envelope of family of curves?

Envelope, in mathematics, a curve that is tangential to each one of a family of curves in a plane or, in three dimensions, a surface that is tangent to each one of a family of surfaces. For example, two parallel lines are the envelope of the family of circles of the same radius having centres on a straight line.

What is the equation of the family of curves orthogonal to the Family Y ax3?

y = ax3, x2 + 3y2 = b. Given a family of curves, we can get an orthogonal trajectory of the family by solving the differential equation.

What do you mean by orthogonal trajectory of a family of curves?

Any curve, which cuts every member of a given family of curves at right angles, is called an orthogonal trajectory of the family. It is not necessary that the curve should intersect every member of the family.

What is a one parameter family?

In this context, the arbitrary constant is called a parameter and the family of solutions is called a one-parameter family. Remark In calculus you learned that not only is each member of the family y = F(x) + C a. solution of the differential equation but this family actually represents the set of all solutions of the.

What is envelope in calculus?

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