KC Sinha Mathematics Solution Class 12 Chapter 12 द्वितीय कोटि का अवकलज (Second Order Derivative) Exercise 12.1 (Q10-Q12)

Exercise 12.1

यदि(If) $y=3 e^{2 x}+2 e^{3 x}$ , सिद्द करे कि (prove that) $\frac{d^{2} y}{d x^{2}}-5 \frac{d y}{d x}+6 y=0$
Sol:

$y=3 e^{2 x}+2 e^{3 x}$

Differentiating w.r.t x

$\frac{d y}{d}=3 \cdot e^{2 x} \cdot 2+2 \cdot e^{3 x} \cdot 3$

$\frac{d y}{d x}=6 e^{2 x}+6 \cdot e^{3 x}$

Again ,Differentiating w.r.t x

$\frac{d^{2} y}{d x^{2}}=6 \cdot e^{2 x} \cdot 2+6 \cdot e^{3 x} \cdot 3$

$\frac{d^{2} y}{d x^{2}}=12 e^{2 x}+18 e^{3 x}$

$\frac{d^{2} y}{d x^{2}}=30 e^{2 x}+30 e^{3 x}-18 e^{2 x}-12 e^{3 x}$

$\frac{d^{2} y}{d x^{2}}=5\left(6 e^{2 x}+6 e^{3 x}\right)-6\left(3 e^{2 x}+2 e^{3 x}\right)$

$\frac{d^{2} y}{d x^{2}}=5 \frac{d y}{d x}-6 y$

$\frac{d^{2} y}{d x^{2}}-\frac{5 dy}{dx}+6 y=0$

यदि(If) $y=\mathrm{Ae}^{mx}+\mathrm{Be}^{2x}$ , दिखाएँ कि (show that) $\frac{d^{2} y}{d x^{2}}-(m+n) \frac{d y}{d x}+m n y=0$
Sol :

$y=\mathrm{Ae}^{mx}+\mathrm{Be}^{2x}$

Differentiating w.r.t x

$\frac{d y}{d x}=A e^{m x} \times m+B \cdot e^{n x} \cdot n$

$\frac{d y}{d x}=m A e^{m x}+n B e^{n x}$

Again , Differentiating w.r.t x

$\frac{d^{2} y}{d x^{2}}=m A \cdot e^{m x} \cdot m+n B \cdot e^{n x} \cdot n$

$\frac{d^{2} y}{d x^{2}}=m^{2} A e^{m x}+n^{2} B e^{n x}$

$\frac{d^{2} y}{d x^{2}}=m^{2} A e^{m x}+m n B e^{n x}+m n A e^{m x}+n^2 Be^{nx}-mnAe^{mx}-mnBe^{nx}$

$\frac{d^{2} y}{d x^{2}}=m\left(m A e^{m x}+n B e^{n x}\right)+n\left(m A e^{m}+n B e^{nx}\right)-mn(Ae^{mx}+Be^{nx})$

$\frac{d^{2} y}{d x^{2}}=(m+n) \cdot\left(m A e^{m x}+n B e^{n x}\right)-m n\left(A e^{m x}+B e^{nx}\right)$

$\frac{d^{2} y}{d x^{2}}=(m+n) \frac{d y}{d x}-m n y$

$\frac{d^{2} y}{d x^{2}}-(m+n) \frac{d y}{d}+m n y=0$

यदि(If) $y=500 e^{7 x}+600 e^{-7 x}$ , दिखाएँ कि (show that ) $\frac{d^{2} y}{d x^{2}}=49 y$
Sol :

$y=500 e^{7 x}+600 e^{-7 x}$

Differentiating w.r.t x

$\frac{d y}{d x}=500 e^{7 x} \cdot 7+600 e^{-7 x} \cdot(-7)$

$\frac{d y}{d x}=7 \times 500 e^{7 x}-7 \times 600 e^{-7 x}$

Again , Differentiating w.r.t x

$\frac{d^{2} y}{d x^{2}}=7 \times 500 e^{7 x} \times 7-7 \times 600 e^{-7x} \times(-7)$

$\frac{d^{2}y}{d^{2}x}=49\left[500 e^{7 x}+600 e^{-7x}\right]$

$\frac{d^{2} y}{d x^{2}}=49 y$