Question

Solve the differential equations: y=x(p²-2p+2)

Answer 1

Answer on Question #79091 - Math - Differential Equations

Question

Solve the differential equations: $y = x(p^2 - 2p + 2)$

Solution

$$\frac{d^2 x}{d y^2} - 2 \frac{dx}{dy} + 2x = y$$

CF:

$$\lambda^2 - 2\lambda + 2 = 0$$ $$\lambda_{1,2} = 1 \pm i$$ $$x_{CF} = (A \cos y + B \sin y) e^y$$

PI:

$$x_{PI} = C y + D$$ $$0 - 2C + 2C y + 2D = y$$ $$\left\{ \begin{array}{l} 2C = 1 \\ 2D - 2C = 0 \end{array} \right. \Rightarrow \left\{ \begin{array}{l} C = 1/2 \\ D = 1/2 \end{array} \right.$$ $$x_{PI} = \frac{y + 1}{2}$$ $$x = x_{CF} + x_{PI} = (A \cos y + B \sin y) e^y + \frac{y + 1}{2}$$

Answer: $x = (A \cos y + B \sin y) e^y + \frac{y + 1}{2}$

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