Key to "Give the correct answer and solve the given..."

sjeikdom0

Answered question

2021-01-08

Give the correct answer and solve the given equation

x y'' - y' = (3 + x) x^{2} e^{x}

Answer & Explanation

krolaniaN

Skilled2021-01-09Added 86 answers

Given: consider the differential equation
$x y " - y^{'} = (3 + x) x^{2} e^{x}$
$\Rightarrow \frac{x y - y^{'}}{x^{2}} = (3 + x) e^{x}$
$\Rightarrow d (\frac{y^{'}}{x}) = (2 + x) e^{x} d x$
$\Rightarrow \int d (\frac{y^{'}}{x}) = \int (3 + x) e^{x} d x + C$
C is the integrating constant
$\Rightarrow \frac{y^{'}}{x} = 3 e^{x} + x e^{x} - e^{x} +$
$\Rightarrow y^{'} = 2 x e^{x} + x 62 e^{x} + C x$
$\Rightarrow \frac{d y}{d x} = 2 x e^{x} + x^{2} e^{x} + C x$
$\Rightarrow \int d y = 2 \int x e^{x} d x + \int x^{2} e^{x} d x + C \int x d x + C_{2}$
$C_{2}$ is the integrating constant
$\Rightarrow y = 2 (x e^{x} - e^{x}) + x^{2} e^{x} - 2 x e^{x} + x e 6 x + C \frac{x^{2}}{2} + C_{2}$
$\Rightarrow$ $y = x^{2} e^{x} + X_{1} x^{2} + c_{2}$
Where $c_{1} = \frac{C}{2}$ is the integrating constant
Finally result:
$\Rightarrow y = x^{2} e^{x} + X_{1} x^{2} + c_{2}$

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