A differential equation of the form y' + p(x)y + q(x)y = r(r) is called a...

  

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A differential equation of the form y' + p(x)y + q(x)y² = r(r) is called a Riccati equations. a) If y = Y(r) is a known solution to the Riccati equation, show that the substitution y = Y(z) + v-¹(x) reduces it to the linear equation v'- [p(x) + 2Y(x)q(x)]v=q(x) b) Find the general solution to the Riccati equation x²y - xy - x²y² = 1, x > 0, given that y = -¹ is a solution. A differential equation of the form y' + p(x)y + q(x)y² = r(r) is called a Riccati equations. a) If y = Y(r) is a known solution to the Riccati equation, show that the substitution y = Y(z) + v-¹(x) reduces it to the linear equation v'- [p(x) + 2Y(x)q(x)]v=q(x) b) Find the general solution to the Riccati equation x²y - xy - x²y² = 1, x > 0, given that y = -¹ is a solution.

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To show that the substitution y Yx v reduces the Riccati equation to a linear equation we need to su... View the full answer