Differential Equations homework problem.

[tarmon.gaidon]

Homework Statement

The equation $$\frac{dy}{dx}$$ = A(x)y²+B(x)y+C(x) is called a Riccati equation. Suppose that one particular solution y₁(x) of this equation is known. Show that the substitution

y = y₁+$$\frac{1}{v}$$

transforms the Riccati equation into the linear equation

$$\frac{dv}{dx}$$+ (B+2Ay₁)v = -A.

The Attempt at a Solution

So i know that y' = y₁' - $$\frac{v'}{v^2}$$ and I have tried substituting y and y' back into the original equation in order to simplify it down to the linear equation given. Unfortunately for some reason I am not getting anywhere. Any help would be appreciated!

Thanks,
Rob

P.S. Sorry for the crappy formatting, not really sure what I am doing.

 

[Mark44]

Homework Statement

The equation $$\frac{dy}{dx}$$ = A(x)y²+B(x)y+C(x) is called a Riccati equation. Suppose that one particular solution y₁(x) of this equation is known. Show that the substitution

y = y₁+$$\frac{1}{v}$$

transforms the Riccati equation into the linear equation

$$\frac{dv}{dx}$$+ (B+2Ay₁)v = -A.

The Attempt at a Solution

So i know that y' = y₁' - $$\frac{v'}{v^2}$$ and I have tried substituting y and y' back into the original equation in order to simplify it down to the linear equation given. Unfortunately for some reason I am not getting anywhere. Any help would be appreciated!

Thanks,
Rob

P.S. Sorry for the crappy formatting, not really sure what I am doing.

When you make your substitution, the left side will be dy₁/dx - v'/v. Are you remembering to replace dy₁/dx by A(x)y₁² + B(x)y₁ + C(x)?