Question

Solve the partial differential equation pq = x.​

Answer 1

Answer:

solve the partial differential equation pq=×

mark brillient

Answer 2

Answer:

The partial differential equation pq = x is F(p(x, y), q(x, y)) = C

where F is an arbitrary function and C is an arbitrary constant.

Step-by-step explanation:

The partial differential equation pq = x can be solved using the method of characteristics. The method of characteristics involves finding the solution of the equation along characteristic curves, which are curves along which the solution is constant.

Here's one way to solve the equation:

Let x(t) and y(t) be the characteristic curves, where t is a parameter.

Along the characteristic curves, p and q are functions of x and y, so we have:

p = p(x(t), y(t))

q = q(x(t), y(t))

Substituting p and q into the equation pq = x, we get:

p(x(t), y(t)) * q(x(t), y(t)) = x(t)

To find x(t) and y(t), we need to find the derivatives of x and y w.r.t. t.

Taking the derivative of both sides w.r.t. t, we get:

p(x(t), y(t)) * q'(x(t), y(t)) + q(x(t), y(t)) * p'(x(t), y(t)) = x'(t)

Setting x'(t) = q(x(t), y(t)), we get:

p'(x(t), y(t)) = x(t) / q(x(t), y(t))

Similarly, setting y'(t) = -p(x(t), y(t)), we get:

q'(x(t), y(t)) = -x(t) / p(x(t), y(t))

Integrating these equations, we get:

p(x, y) = C1 / q(x, y)

q(x, y) = -C2 / p(x, y)

where C1 and C2 are arbitrary constants.

Solving for p and q in terms of x and y, we get:

p(x, y) = sqrt(x / C2)

q(x, y) = -sqrt(-x / C1)

The general solution of the partial differential equation pq = x is given by:

F(p(x, y), q(x, y)) = C

where F is an arbitrary function and C is an arbitrary constant.

To learn more partial differential equations, click on the link below:

https://brainly.in/question/50633293

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https://brainly.in/question/51680704

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