solve 2y^2(y'')+2y(y')^2=1
To solve the given differential equation, we can apply the method of separable variables.
Step 1: Rewrite the equation in standard form.
The given equation is: 2y^2(y'') + 2y(y')^2 = 1
Step 2: Factor out common terms.
We can rearrange the equation as follows:
2y(y'') + 2y(y')^2/y = 1/y^2
Step 3: Simplify the equation.
Divide both sides of the equation by 2y to simplify it further:
(y'') + (y')^2/y = 1/(2y^3)
Step 4: Separate the variables.
Let's substitute u = y' to simplify the equation.
Differentiating u with respect to x, we get:
u' = (d/dx)(y') = y''
Now, the equation becomes:
u' + u^2/y = 1/(2y^3)
Step 5: Solve the resulting ordinary differential equation.
The equation is separable, so we can rewrite it as:
u' = 1/(2y^3) - u^2/y
Next, multiply both sides by y/u^2 to separate the variables:
(y/u^2)du = 1/(2y^3) - 1/u^3 dy
Integrating both sides of the equation gives us:
∫(y/u^2)du = ∫(1/(2y^3) - 1/u^3)dy
Integrating each term separately, we obtain:
(1/u) + (1/2y^2) = (-1/2y^2) + (1/u^2) + C
where C is the constant of integration.
Step 6: Rearrange the equation to solve for u.
Combining like terms, we get:
(1/u) + (1/2y^2) = (1/u^2) + C
Multiplying through by 2u^2y^2, we obtain:
2y^2 + u = 2y^2 u^2 + Cu^2
Rearranging the equation gives us a quadratic equation in terms of u:
2y^2 u^2 - u + Cu^2 - 2y^2 = 0
Step 7: Solve for u using the quadratic formula.
Applying the quadratic formula, we have:
u = (-(-1) ± sqrt((-1)^2 - 4(2y^2)(Cu^2 - 2y^2))) / (2(2y^2))
Simplifying further, we get:
u = (1 ± sqrt(1 + 8y^2(Cu^2 - 2y^2))) / (4y^2)
Step 8: Solve for y'.
Since y' = u, we can substitute u back into the equation:
y' = (1 ± sqrt(1 + 8y^2(C(y')^2 - 2y^2))) / (4y^2)
The resulting equation is a first-order nonlinear ordinary differential equation for y'. Further simplification or analytic solution of this equation is not possible in this case.
In summary, we have obtained an equation in the form y' = f(y), which is implicit in nature. The solution to this equation can be obtained numerically or through specialized techniques for solving nonlinear differential equations, such as using a computer program or numerical methods like Euler's method or the Runge-Kutta method.