“Solving a differential equation” is also referred to as “integrating a differential equation”, since the process of finding the solution to a differential equation involves integration.

**Solution of First
Order and First Degree Differential Equations**

**Variables
Separable Method**

In
solving differential equations, separation of variables was introduced
initially by Leibniz and later it was formulated by John Bernoulli in the year
1694.

A first
order differential equation is separable if it can be written as *h* ( *y*)
*y* ′ = *g*(
*x*) where the left side is a product
of *y*′ and a function of *y* and the right side is a function of *x* . Rewriting a separable differential
equation in this form is called the method of separation of variables.

Finding
a solution to a first order differential equation will be simple if the
variables in the equation can be separated. An equation of the form *f*_{1} (*x*)*g*_{1} ( *y*)*dx*
+
*f* _{2} (*x*)*g* _{2} ( *y*)*dy*
=
0 is called an equation with **variable separable** or simply a **separable
equation**.

Rewrite
the given differential equation as

Integration
of both sides of (1) yields the general solution of the given differential
equation as , where *C* is an arbitrary constant.

**Remarks**

1. No
need to add arbitrary constants on both sides as the two arbitrary constants
are combined together as a single arbitrary constant.

2. A
solution with this arbitrary constant is the general solution of the
differential equation.

“Solving a differential equation” is also referred to as
“integrating a differential equation”, since the process of finding the
solution to a differential equation involves integration.

** **

**Example 10.11**

** **

**Example 10.12**

Find the
particular solution of (1+ *x*^{3}
)*dy* − *x*^{2}
*ydx* = 0 satisfying the condition *y*(1) = 2

**Solution**

Given
that (1+
*x*^{3} )*dy* − *x*^{2}*ydx* = 0 .

The above equation is written as

Hence, *y *^{3}* *=* C*(1+* x*^{3}* *) gives the general solution of the given differential
equation. It is given that when *x *=* *1,* y *=* *2. Then 2^{3} = *C*(1+1)
⇒* C *=* *4* *and hence the
particular solution is *y*^{3}* *=* *4(1+* x*^{3})*
*.

Tags : Solution of First Order and First Degree Differential Equations | Mathematics , 12th Maths : UNIT 10 : Ordinary Differential Equations

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12th Maths : UNIT 10 : Ordinary Differential Equations : Variables Separable Method | Solution of First Order and First Degree Differential Equations | Mathematics

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