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**Hint:**To find the given equation is a separable equation in which to solve this differential equation we need to integrate both sides of the equation.

**Complete step by step answer:**

The given differential equation is

\[\dfrac{{dy}}{{dx}} = \dfrac{{3{x^2}}}{{{e^{2y}}}}\] ………………………. 1

As the given equation is separable equation, multiply the terms with respect to x and y as

\[{e^{2y}}dy = 3{x^2}dx\]

Hence to solve this differential equation we need to integrate both sides of the equation.

\[\int {{e^{2y}}dy} = \int {3{x^2}dx} \]

The integration of \[{e^{2y}}\] is \[\dfrac{1}{2}{e^{2y}}\] and \[3{x^2}\]is \[{x^3} + c\], hence we get

\[\dfrac{1}{2}{e^{2y}} = {x^3} + c\]…………………… 2

Here we need to find the value of c, hence let us use the given function for \[f\left( 0 \right) =

\dfrac{1}{2}\]in the obtained equation i.e., equation 2 as

\[\dfrac{1}{2}{e^{2\left( {\dfrac{1}{2}} \right)}} = {0^3} + c\]

Simplifying the equation, the value c is

\[c = \dfrac{1}{2}e\]

Hence, equation 2 after substituting the value of c is

\[\dfrac{1}{2}{e^{2y}} = {x^3} + c\]

\[\dfrac{1}{2}{e^{2y}} = {x^3} + \dfrac{1}{2}e\]

Simplifying the terms, we get

\[{e^{2y}} = 2{x^3} + e\]

\[2y = \ln \left( {2{x^3} + e} \right)\]

**Therefore, the value of \[y\] is**

\[y = \dfrac{1}{2}\ln \left( {2{x^3} + e} \right)\]

\[y = \dfrac{1}{2}\ln \left( {2{x^3} + e} \right)\]

**Additional information:**

A differential equation is an equation which contains one or more terms and the derivatives of one variable (i.e., dependent variable) with respect to the other variable (i.e., independent variable).

\[\dfrac{{dy}}{{dx}} = f\left( x \right)\]

Here \[x\] is an independent variable and \[y\] is a dependent variable

The order of the highest order derivative present in the differential equation is called the order of the equation. If the order of the differential equation is 1, then it is called first order. If the order of the equation is 2, then it is called a second-order, and so on.

**Note:**The key point to find the differential equation is we need to find the order of derivative i.e., its highest order derivative present in the differential equation, based on that we can differentiate or apply integral to the equation. The primary purpose of the differential equation is the study of solutions that satisfy the equations and the properties of the solutions. One of the easiest ways to solve the differential equation by using explicit formulas.

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