Question

(a) The differential equation (2x2 + by2)dx + cxy dy = ( can made exact by multiplying with integrating factor 1 1 Then find x2- the relation between b and c​

Answer 1

Answer:

please mark brillant answer

Step-by-step explanation:

Multiplying the differential equation by 1/x2, we get

It is exact

So,

implies

2b + c = 0

Answer 2

The relation between b and c​ is c = -2b.

• Specifically, an exact equation is a differential equation that can be solved instantly without the aid of any specialised methods. If the result of a straightforward differentiation, a first-order differential equation (of one variable) is referred to as an exact differential or exact differential.
• The equation R(x, y) = c (where c is constant) will implicitly define a function y that will fulfil the original differential equation. This function is R(x, y), the partial x-derivative of which is Q, and the partial y-derivative of which is P.

Here, according to the given information, we are given that,

$(2x^{2} + by^{2} )dx + cxy dy = 0$

Multiplying by integrating factor which is $\frac{1}{x^{2} }$, we get,

$(2+b\frac{y^{2} }{x^{2} } )dx+c\frac{y}{x} dy=0$

Or, $-c\frac{y}{x^{2} } =2b\frac{y}{x^{2} }$

Or, c = -2b

Hence, the relation between b and c​ is c = -2b.

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