Exercise 12.1
Question 16
यदि(If) y=e^{-x} \cos x , दिखाएँ कि (show that) \frac{d^{4} y}{d x^{4}}+4 y=0Sol :
y=e^{-x} \cos x
Differentiating w.r.t x
\frac{d{y}}{d{x}}=e^{-x} \times(-1) \cdot \cos x+e^{-x} \times(-\sin x)
\frac{d y}{d x}=-e^{-x} \cos x-e^{-x} \sin x
\frac{d y}{d x}=-e^{-x}[\cos x+\sin x]
Again , Differentiating w.r.t x
\frac{d^2 y}{d x^{2}}=-e^{-x}(-1)(\cos x+\sin x)+\left(-e^{-x}\right)(-\sin x+\cos x)
\frac{d^{2} y}{d x^{2}}=e^{-x}[\cos x+\sin x+\sin x-\cos x]
\frac{d^{2} y}{d x^{2}}=2 e^{-x} \sin x
Again , Differentiating w.r.t x
\frac{d^{3} y}{d x^{3}}=2\left[e^{-x} \cdot(-1) \sin x+e^{-x} \cos x\right]
\frac{d^{3} y}{d x^{3}}=2 e^{-x}(-\sin x+\cos x)
Again , Differentiating w.r.t x
\frac{d^{4} y}{d x^{4}}=2\left[e^{-x} \times(-1)(-\sin x+\cos x)+e^{-x}(-\cos x-\sin x)\right]
\frac{d^{4} y}{d x^{4}}=2 e^{-x}[\sin x-\cos x-\cos x-\sin x]
\frac{d^{4} y}{d x^{4}}=2 \cdot e^{-x} \times(-2 \cos x)
\frac{d^{4} y}{d x^{4}}=-4 e^{-x} \cos x
\frac{d^{4} y}{d x^{4}}=-4 y
\frac{d^{4} y}{d x^{4}}+4 y=0
Question 17
यदि (If) e^{y}(x+1)=1 , दिखाएँ कि (show that) \frac{d^{2} y}{d x^{2}}=\left(\frac{d y}{d x}\right)^{2}Sol :
e^{y}(x+1)=1
Differentiating w.r.t x
e^{y} \cdot \frac{d y}{dx}(x+1)+e^{y} \cdot 1=0
e^{y}\left[(x+1) \cdot \frac{d y}{d x}+1\right]=0
(x+1) \frac{d y}{dx}+1=0
\left[ \frac{d y}{d x}=\frac{-1}{x+1} \right]
Again , Differentiating w.r.t x
1 \cdot \frac{d y}{d x}+(x+1) \cdot \frac{d^{2} y}{d x^{2}}+0=0
(x+1) \frac{d^{2} y}{d x^{2}}=-\frac{d y}{d x}
\frac{d^{2} y}{d x^{2}}=-\frac{1}{(x+1)} \cdot \frac{d y}{d x}
\frac{d^{2} y}{dx^{2}}=\frac{d y}{dx} \cdot \frac{dy}{dx}
\frac{d^{2} y}{d x^{2}}=\left(\frac{d y}{d x}\right)^{2}
Question 18
यदि(If) y=e^{a \cos ^{-1} x} , -1≤x≤1 दिखाएँ कि (show that) \left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-a^{2} y=0Sol :
y=e^{a \cos ^{-1} x}
Differentiating w.r.t x
\frac{d y}{d x}=e^{a \cos ^{-1} x} \times a \cdot \frac{-1}{\sqrt{1-x^{2}}}
\frac{d y}{d x}=\frac{-a e^{a \cos^{-1}}x}{\sqrt{1-x^{2}}}
Again , Differentiating w.r.t x
\frac{d^{2} y}{d x^{2}}=\frac{-a \cdot e^{a \cos ^{-1} x}.a \cdot \frac{1}{\sqrt{1-x^{2}}}\times \sqrt{1-x^{2}}-\left.(-a e^{\left.a \cos ^{-1} x\right.}) \times \frac{1}{2 \sqrt{1-x^{2}}}\right.\times -2x}{(\sqrt{1-x^{2}})^{2}}
\frac{d^{2} y}{d x^{2}}=\frac{a^{2} e^{a \cos^{-1}}x-\frac{a x \cdot e^{a \cos^{-1}}x}{\sqrt{1-x^{2}}}}{1-x^{2}}
\left(1-x^{2}\right) \cdot \frac{d^{2} y}{d x^{2}}=a^{2} \cdot y+x \cdot \frac{d y}{d x}
\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-a^{2} y=0
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