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Consider the differential equation

$$\frac{\mathrm{d}y}{\mathrm{d}x} -\frac{3y}{x} =x^3\cos(x) (x>0)$$

Find the general solution expressing $y$ explicitly as a function of $x$.

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    Use the integrating factor method. – 2017-02-22
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    the solution containes the Bessel function – 2017-02-22
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    sorry what i have wrote is not true, i have had $y''$ enstead of $y'$ – 2017-02-22

1 Answers 1

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  1. The general solution of the hom. equation $\frac{\mathrm{d}y}{\mathrm{d}x} -\frac{3y}{x} =0$ is given by

$y_h(x)=cx^3$ ($c \in \mathbb R$).

  1. A spcial solution $y_s$ of the inhom. eqaution you get from

$y_s(x)=C(x)x^3$. Determine the function $C$.

(Controll: $C(x)= \sin x$)

  1. The solutions of $\frac{\mathrm{d}y}{\mathrm{d}x} -\frac{3y}{x} =x^3\cos(x)$ are then given by

$y(x)=cx^3+x^3 \sin x$ ($c \in \mathbb R$).