For equation ` 6x-5y=8`, verify that (3,2) is a solution.

Verify that the function y = e^(-3x) , is a solution of the differential equation ((d^2y)/dx^2) + (dy)/dx - 6y = 0

Solve the equation (x)/(2)-6=8-(2x)/(3) and verify the result.

Find the solutions of form x = a, y = 0 and x= 0, y= b for the following pairs of equations. Do they have any common such solution ? 7x+3y=42 and 2x+5y=12 .

The equation 2x + 5y = 7 has a unique solution, if x, y are :

Show that the system of eqations 3x - y + 4z = 3, x + 2y - 3z = -2, 6x + 5y + lambda z = -3 has atleast one solution for every real lambda , Find the set of solutions when lambda = -5

A person standing at the junction (crossing) of two straight paths represented by the equations 2x -3y + 4 = 0 and 3x + 4y -5 = 0 wants to reach the path whose equation is 6x -7y + 8 = 0 in the least time. Find equation of the path that he should follow.

Verify that the given function (implicit or explicit) is a solution of the corresponding differential equation: y = xsin3x : ((d^2y)/(dx^2)) + 9y - 6cos3x = 0

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: y = x sinx : xy' = y + x sqrt(x^2 - y^2)

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: y = x^2 + 2x + C : y' - 2x - 2 = 0

Verify that the function y= cos x + c is a solution of the differential equation y' + sin x=0