The integral surface of the p. d. esatisfying the condition u ( 1 , y ) = y is given bya)u(x,y) = y/xb)u(x, y) =c)u(x, y)d)u(x,y) = y + x - 1Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Think of the PDE as ∇u⋅(a(x,y),b(x,y))=0∇u⋅(a(x,y),b(x,y))=0 so that, geometrically it must be that dy/dx = [b(x,y)/a(x,y)] The solution of the ODE defines the characteristic curves of the PDE, along which solutions (to the PDE) are constant. Then u(x,y) = f(C) where the function ff is determined from the given auxiliary condition.

For example, here the ODE is dy/dx = y/x

so C = ln|y/x|

Then u(x,y) = f(C) = f(ln|y/x|).

To determine f, apply the given condition: u(1,y) = f(ln|y|) = y

⟹ f(y) = e^y

Thus, u(x,y) = exp(ln|y/x|)

= |y/x|

Similar Mathematics Doubts

The integral surface satisfying the equationand passing through thecurve x = 1 — t ,y = l + t , z = l + t2 is

Consider the vector fieldand surface integralwhereS be is the surface consists of the threez = 5 - x2- y2, 1 < z < 5 on the top ; x2 + y2 = 1 , 0<z < 1 on the bottom ; z = 0 base. Then the flux of across the surface S is _______ Correct answer is '0'. Can you explain this answer?

If the differential equation is of the type f(D)y = sin ax, where f(D) is a polynomial in D containing the odd powers in D only, then

Let S be the surface of cone z=√x^2 y^2 bounded by the planes z=0 and z=3. further, let C be the closed curve forming the boundary of the surface S. A vector field F is such that curl F =-xi-yj .the absolute value of the line integral F.dr?

Let S be the surface of the cone z=√x² y² bounded by the planes z=0 and z=3. Further let C be the closed surface forming the boundary of the surface S. A vector field is such that curl F is -xi-yj. Find absolute value of line integral F.dr?

Top Courses for Mathematics

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams