# Find an explicit solution of the initial value problem

Find an explicit solution of the initial value problem $$dy/dx = x^4 / (y+1)$$ when $y(1) = 2$

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## 1 Answer

$$\dfrac{dy}{dx} = \dfrac{x^4}{y+1} \implies (y+1) dy = x^4 dx \implies \dfrac{(y+1)^2}2 = \dfrac{x^5}5 + c$$ When $x=1$, $y=2$ and hence $$\dfrac{(2+1)^2}2 = \dfrac{1}5 + c \implies c = \dfrac92 - \dfrac15 = \dfrac{43}{10}$$ Hence, $$\dfrac{(y+1)^2}2 = \dfrac{x^5}5 + \dfrac{43}{10}$$

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