Solve the given differential equation x2y 3xy 6y 0Solutionx

Solve the given differential equation. x2y\'\' 3xy\' 6y = 0

Solution

x²y\'\' 3xy\' 6y = 0

Assume solution proportional to x

==> y\' = x^-1 , y\'\' = ( - 1)x^-2

==> x²(( - 1)x^-2) - 3x(x^-1) - 6(x) = 0

==> ( - 1) x - 3x - 6x = 0

==> x[ ² - - 3 - 6 ] = 0

==> x(² - 4 - 6) = 0

x is never zero for x 0

==> ² - 4 - 6 = 0

==> ² - 2(2) + 2² - 2² = 6

==> ( - 2)² = 6 + 4

==> ( - 2)² = 10

==> - 2 = 10 , -10

==> = 2 + 10 , 2 - 10

for = 2 + 10 , y₁(x) = c₁ x^{2 + 10}

for = 2 - 10 , y₂(x) = c₂ x^{2 - 10}

Hence solution is y(x) = c₁ x^{2 + 10} + c₂ x^{2 - 10}