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Differential Equations

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Moderate

Converting a Nonlinear Equation to a Linear Equation

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Consider the equation $y'(x)=\frac{1}{2y}(x^2+y^2)$. This is a nonlinear equation.

However, if we do a substitution $u=f(y)$ for a suitable $f$, we can transform the equation into a linear equation.

Suppose we require $f(1)=1$, then $f(y)$ is given by: