Ncert class 12 maths differential equation practice problems

 Linear Differential Equations

Definition

An equation containing a variable, its derivative and a few more functions of degree one is called a linear differential equation. The standard form of representing a linear differential equation is dy / dx + Py = Q. In this equation, x is an independent variable, and y is the dependent variable. We find derivatives of dependent variables only with respect to an independent variable. 

General solution formula of a linear differential equation

The general solution of the differential equation dy / x + Py = Q is given by:

y . (I.F) = ∫(Q . (I.F) . dx) + Cy . (I.F) = ∫(Q . (I.F) . dx) + C

Here, I.F. is the integrating factor and is given by: e^ ∫P . dx

Homogeneous differential equation

 A function f(x, y) in x and y is said to be a homogeneous function if the degree of each term in the function is constant (say p). 

In general, a homogeneous function ƒ(x, y) of degree n is expressible as:

ƒ(x, y) = λ^n ƒ(y/x)

Homogeneous differential equation can be expressible as 

dy/dx = f(x,y)

We can solve a homogeneous differential equation of the form dx/dy = f(x, y) where, f(x, y) is a homogeneous function, by simply replacing x/y to v or putting y = vx. Then after solving the differential equation, we put back the value of v to get the final solution. The detailed step for solving the Homogeneous Differential Equation i.e., dy/dx = y/x.

Step 1: Put y = vx in the given differential equation.

Now, if y = vx

then, dy/dx = v + xdv/dx

Substituting these values in the given equation 

Step 2: Simplify and then separate the independent variable and the differentiation variable on either side of the equal to.

v + xdv/dx = vx/x

⇒ v + xdv/dx = v

⇒ xdv/dx = 0

⇒ dv = 0

Step 3: Integrate the differential equation so obtained and find the general solution in v and x.

Integrating both sides,

∫dv = 0

⇒ v = c

Step 4: Put back the value of v to get the final solution in x and y.

Substituting y/x = v

⇒ y/x = c

⇒ y = cx

This is the required solution of the given homogeneous differential equation

Solved problems of the above two types of differential equation 

                        Continued.....