## 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:

- Construct the differential equation G(x,y,y′)=0 for the given family of curves g(x,y)=C.
- Replace y′ with (−1y′) in this differential equation.
- 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?**