Question

i) Solve : (4x2y - 6)dx + x3 dy = 0​

Answer 1

Answer:

dy/dx = (-4x²y - 6)/x³

Step-by-step explanation:

(4x²y - 6)dx + x³ dy = 0​

x³ dy = -(4x²y - 6)dx

dy/dx = (-4x²y - 6)/x³

Answer 2

Answer:

The general solution is x²(x²y-3) = c.

Step-by-step explanation:

Differentiation:

• Differentiation is defined as the rate of change.
• Derivative of independent variable w.r.t dependent variable is also called as differentiation.
• Suppose 's' is the distance travelled then the derivative of s w.r.t time 't' is given by ds/dt.
• The rate of change of distance w.r.t time is known as velosity.
• ds/dt is the velocity.

Finding solution:

Given equation is (4x²y-6)dx+x³dy = 0

The given equation is a Differential equation which is of the form   Mdx+Ndy = 0

         where M = 4x²y-6 and N = x³

Check for Exact D.E:

 The condition for a D.E to be exact is

    (partial derivative of M w.r.t y) = (partial derivative of N w.r.t x)

consider

partial derivative of M w.r.t y = 4x²

consider

partial derivative of N w.r.t x = 3x²

since,            4x² ≠ 3x²

  So, the give D.E is not exact

 To make the given D.E exact

we need to find the Integrating factor.

 To find Integrating Factor(I.F):

we need to check the function

(1/N)[(partial derivative of M w.r.t y)-(partial derivative of N w.r.t x)]

 = {(1/ x³)[4x² -3x²]}

 =[(1/ x³)(x²)]

 = (1/x)

 = f(x)  

     Integrating factor = I.F = e^(∫f(x)dx)

                                           = e^(∫(1/x)dx)

                                           = e^(logx)

                                           = x

multiply the given equation with I.F = x

    x((4x²y-6)dx+x³dy) = x(0)

      (4x³y-6x)dx+x⁴dy = 0

  This is in the form of M₁dx+N₁dy = 0

 where M₁ = 4x³y-6x and N₁ = x⁴

Now check whether the above D.E is exact or not

   (partial derivative of M₁ w.r.t y) = 4x³

   (partial derivative of N₁ w.r.t x) = 4x³

      Since,

    (partial derivative of M₁ w.r.t y) = (partial derivative of N₁ w.r.t x)

      The general solution of the D.E is

       ∫M₁(treating y as constant) dx + ∫ N₁(terms not containing x) dy = c

         ∫(4x³y-6x) dx + ∫(0)dy = c

          ∫(4x³y-6x) dx = c

           4y(x⁴/4)-6(x²/2) = c

              yx⁴ - 3x² = c

              x²(x²y-3) = c

   Hence, the general solution is x²(x²y-3) = c.

         

 

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