Differential Equation Class 12 Notes - Chapter 9

What are Differential Equations?

The equation which involves the derivative of the variable which is dependent with respect to the independent variable is known as differential equations. The order here is given to the highest order of the occurring derivative. The degree is given to the presence of the polynomial equation in the derivatives.

What are Solutions?

The function which is able to satisfy the given differential equation is known as the solution. A general solution is referred to the presence of many arbitrary constants as their order and if the solution is free from these arbitrary constants then it is called as particular solutions. In order to form a differential equation from a function which is given, the function should be differentiated several times as the number of arbitrary constants present in the function and then eliminating them at last.

Variable Separation Method

This type of method is used to solve a certain type of equations in which the variables can be separated entirely which means the terms which contain y should have with dy and the terms which contain x should have with dx.

Homogenous Differential Equation

When the zero degree homogenous functions can be expressed in the form, dy/dx=f(x,y) or dx/dy=g(x,y) where, f(x,y) and g(x,y) are the mentioned functions then it is called homogenous differential equations.

First Order Linear Differential Equations

If R and S are the constants or function of x only and the differential equation is of the form dy/dx+Ry=S.

 

Important Questions

  1. What will be the population of the village in 2019, if the population of the village was 30, 000 in 1888 and 20000 in the year 2003? The rate of increase of the population is continuously at the rate proportional to the number of its inhabitants present at any time.
  2. Find the equation of the curve passing through the point (0,π/6) and differential equation is sin y cos z dx + cos y sin z dy = 0.
  3. If in the first quadrant, a family of circles touches the coordinate axes. Form the differential equation for it.
  4. If the slope of the tangent to the curve at any point (p, q) is equal to the sum of the coordinates of the point then find the equation of a curve passing through the origin.
  5. If the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5 then find the equation of a curve passing through the point (0, 5) given that.

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