Differential equation can be defined as the finding out the differentiation of the mathematicsgiven equation. To find out the differential equation of the function we need to use differential equation formulas. There are two types of differentiation they are ordinary differentiation and partial differentiation. In ordinary differentiation we will differentiate the given function and all the variables with respect to one variable where as in partial differentiation, differentiation is done only with one variable.
Let us discuss some of the problems on differential equations.
Consider the first example to be: find out the differential equation of the given function y equal to x^2 + 2x + 3.
The given function is y equal to x^2 + 2x + 3. Now let us differentiate the given function with respect to x then we will get it as dy / dx equal to d / dx(x^2 + 2x + 3). If we simplify this then we will get it as dy / dx = d /dx (x^2) + d / dx(2x) + d / dx(3) now we know the formula that d / dx (x^n) equal ton x^(n – 1) and differentiation of constant is 0. Now substitute these formulas into the equation then we will get it as dy / dx = 2x + 2 (1) + 0 which is equal to 2x + 2. Now the required differential equation of the given function is dy / dx which is equal to 2x + 2.
Consider the second example to be: find out the differential equation of the function y equal to x^3 + 2x^2 + 4x + 6.
Now the given function is y equal to x^3 + 2x^2 + 4x + 6. Now differentiate the given equation with respect to x then we will get it as dy / dx equal to d / dx (x^3 + 2x^2 + 4x + 6) which is equal to dy / dx equal to d /dx (x^3) + d / dx (2x^2) + d / dx (4x) + d / dx (6). Now use the differentiation formula that is d / dx(x^n) is equal to n x^ (n – 1) and the differentiation of a constant is zero. Now we will get the required differentiation is dy / dx equal to 3 x^ (3 – 1) + 2 ( 2 x^ (2 – 1) ) + 4 (1) + 0. If we simplify this then we will get it as dy / dx equal to 3x^2 + 2( 2x) + 4. Now the required differential equation of the given function that is y which is equal tox^3 + 2x^2 + 4x + 6 is dy / dx which is equal to3x^2 + 4x + 4. This is how the different differential equations can be solved using the properties which re shown above. We should follow the same process as shown in the examples above. These properties helps in solving different types of differential equations.
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