Home > Calculus > Solving Separable Differential Equations

What is the general solution of the differential equation? : # y' = x(1+y^2) #

Answer 1

Emma Aldridge

# y = tan(1/2x^2 + C) #

We have:

# y' = x(1+y^2) #

This can be rewritten as follows: This is a First Order Separable Ordinary Differential Equation.

# 1/(1+y^2) dy/dx = x #

Consequently, we can "separate the variables" to obtain:

# int \ 1/(1+y^2) \ dy = int \ x \ dx #

Since both integrals are typical calculus outcomes, integrating yields:

# arctany = 1/2x^2 + C #

Getting to the Overall Solution:

# y = tan(1/2x^2 + C) #

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Charles Anctil

The general solution of the differential equation ( y' = x(1+y^2) ) is:

[ y = \tan\left(\frac{x^2}{2} + C\right) ]

where ( C ) is the constant of integration.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.