Solving the Differential Equation ( frac{dy}{dx} y - 4x^2 ) using ( y - 4x 0 )

In this article, we will explore how to solve the differential equation ( frac{dy}{dx} y - 4x^2 ) using the substitution ( y - 4x 0 ). This method involves a series of steps including substitution, differentiation, and integration to reach an implicit solution. Let's break down the process step by step.

Substitution

To begin, we use the substitution ( y - 4x v ). This implies that:

[ y v 4x ]

Step 1: Differentiation

Next, we differentiate ( y ) with respect to ( x ) using the chain rule:

[ frac{dy}{dx} frac{dv}{dx} 4 ]

Step 2: Rewrite the equation

Substitute ( frac{dy}{dx} ) and ( y ) into the original differential equation:

[ frac{dv}{dx} 4 (v 4x) - 4x^2 ]

Simplifying the right-hand side:

[ frac{dv}{dx} 4 v - 4x^2 4x ]

Subtracting 4 from both sides:

[ frac{dv}{dx} v - 4x^2 4x - 4 ]

Step 3: Separate variables

We can rewrite the equation as:

[ frac{dv}{dx} v - 4(v^2 - 4x^2 4x) - 4 ]

Rearranging terms:

[ frac{dv}{dx} v^2 - 4 ]

Now, separate the variables:

[ frac{dv}{v^2 - 4} dx ]

Step 4: Partial fractions

Decompose the left side using partial fractions:

[ frac{1}{v^2 - 4} frac{A}{v - 2} frac{B}{v 2} ]

Solving for ( A ) and ( B ):

[ 1 A(v 2) B(v - 2) ]

Setting ( v 2 ):

[ 1 4A quad Rightarrow quad A frac{1}{4} ]

Setting ( v -2 ):

[ 1 -4B quad Rightarrow quad B -frac{1}{4} ]

Therefore:

[ frac{1}{v^2 - 4} frac{1}{4} left( frac{1}{v - 2} - frac{1}{v 2} right) ]

Step 5: Integrate both sides

Integrate both sides of the equation:

[ int frac{1}{4} left( frac{1}{v - 2} - frac{1}{v 2} right) dv int dx ]

Simplifying the integral:

[ frac{1}{4} left( ln|v - 2| - ln|v 2| right) x C ]

Combining the logarithms:

[ frac{1}{4} ln left| frac{v - 2}{v 2} right| x C ]

Multiplying both sides by 4:

[ ln left| frac{v - 2}{v 2} right| 4x 4C ]

Step 6: Back-substitution

Substitute ( v y - 4x ) back into the equation:

[ ln left| frac{(y - 4x) - 2}{(y - 4x) 2} right| 4x 4C ]

Let ( K e^{4C} ), the final implicit solution is:

[ left| frac{y - 4x - 2}{y - 4x 2} right| Ke^{4x} ]

Summary

The solution to the differential equation ( frac{dy}{dx} y - 4x^2 ) using the substitution ( y - 4x 0 ) is an implicit relationship between ( y ) and ( x ). Depending on the initial conditions, this relationship can be used to find the explicit form of ( y ) as a function of ( x ).