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Question Number 104509 by bobhans last updated on 22/Jul/20

$$\left(1\right)\frac{d^{2}y}{{dx}^2}-7\frac{dy}{dx}+12y=0$$$$\left(2\right)y^{‴}-3y^{″}+4y^{\prime }-2y=e^x-\cos x$$$$\left(3\right)y^{‴}+3y^{″}+5y^{\prime }+3y=0$$$$\left(4\right)\underset{x\to 0}{\lim }\frac{\cot 4x-1}{2x^2}$$

Answered by bramlex last updated on 22/Jul/20

$$\left(1\right)homogenoussolution$$$${\lambda }^2-7\lambda +12=0;\lambda =3,4$$$$\therefore y=C_1{e}^{3x}+C_2{e}^{4x}$$$$\left(4\right)\underset{x\to 0}{\lim }\frac{1-\tan 4x}{2x^2\tan 4x}=\mathrm{\infty }$$

Answered by Dwaipayan Shikari last updated on 22/Jul/20

$$\underset{x\to 0}{\lim }\frac{1-tan4x}{2x^{2}tan4x}=\frac{1-4x+\frac{64x^3}{3}}{2x^2(x-\frac{x^3}{3})}\to \mathrm{\infty }$$

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