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Q11:
If $y = 5 \cos x - 3 \sin x$, prove that $\frac{d^2y}{dx^2} + y = 0$
Solution :
Step 1: Establishing the Initial Function
The given function is defined as:
$y = 5 \cos x - 3 \sin x$ — (Eq. 1)
Step 2: Calculating the First Derivative
Differentiate $y$ with respect to $x$ using the basic differentiation rules for trigonometric functions, specifically $\frac{d}{dx}(\cos x) = -\sin x$ and $\frac{d}{dx}(\sin x) = \cos x$ [By the Sum/Difference Rule of Differentiation]:
$\frac{dy}{dx} = \frac{d}{dx}(5 \cos x) - \frac{d}{dx}(3 \sin x)$
$\frac{dy}{dx} = 5(-\sin x) - 3(\cos x)$
$\frac{dy}{dx} = -5 \sin x - 3 \cos x$ — (Eq. 2)
Step 3: Calculating the Second Derivative
Differentiate the first derivative $\frac{dy}{dx}$ with respect to $x$ to obtain the second-order derivative $\frac{d^2y}{dx^2}$:
$\frac{d^2y}{dx^2} = \frac{d}{dx}(-5 \sin x - 3 \cos x)$
Applying the linearity of the derivative operator:
$\frac{d^2y}{dx^2} = -5 \frac{d}{dx}(\sin x) - 3 \frac{d}{dx}(\cos x)$
$\frac{d^2y}{dx^2} = -5(\cos x) - 3(-\sin x)$
$\frac{d^2y}{dx^2} = -5 \cos x + 3 \sin x$ — (Eq. 3)
Step 4: Algebraic Verification of the Identity
We are required to prove that $\frac{d^2y}{dx^2} + y = 0$. Substituting Eq. 3 and Eq. 1 into the expression:
$\text{LHS} = \frac{d^2y}{dx^2} + y$
$\text{LHS} = (-5 \cos x + 3 \sin x) + (5 \cos x - 3 \sin x)$
Grouping the terms by trigonometric functions:
$\text{LHS} = (-5 \cos x + 5 \cos x) + (3 \sin x - 3 \sin x)$
$\text{LHS} = 0 + 0 = 0$
$\text{LHS} = \text{RHS}$
Visualization of the Function
Final Solution: Since the substitution of the second derivative and the original function results in zero, the identity $\frac{d^2y}{dx^2} + y = 0$ is proven.
More Questions from Class 12 Mathematics Continuity and Differentiability EXERCISE 5.7
- Q1: Find the second order derivatives of the functions $x^2 + 3x + 2$
- Q2: Find the second order derivatives of the functions $x^{20}$
- Q3: Find the second order derivatives of the functions $x \cdot \cos x$
- Q4: Find the second order derivatives of the functions $\log x$
- Q5: Find the second order derivatives of the functions. $x^3 \log x$
- Q6: Find the second order derivatives of the functions. $e^x \sin 5x$
- Q7: Find the second order derivatives of the functions. $e^{6x} \cos 3x$
- Q8: Find the second order derivatives of the functions. $\tan^{-1} x$
- Q9: Find the second order derivatives of the functions. $\log(\log x)$
- Q10: Find the second order derivatives of the functions. $\sin(\log x)$
- Q12: If $y = \cos^{-1} x$, Find $\frac{d^2y}{dx^2}$ in terms of $y$ alone.
- Q13: If $y = 3 \cos(\log x) + 4 \sin(\log x)$, show that $x^2 y_2 + xy_1 + y = 0$
- Q14: If $y = Ae^{mx} + Be^{nx}$, show that $\frac{d^2y}{dx^2} - (m + n)\frac{dy}{dx} + mny = 0$
- Q15: If $y = 500e^{7x} + 600e^{-7x}$, show that $\frac{d^2y}{dx^2} = 49y$
- Q16: If $e^y (x + 1) = 1$, show that $\frac{d^2y}{dx^2} = \left(\frac{dy}{dx}\right)^2$
- Q17: If $y = (\tan^{-1} x)^2$, show that $(x^2 + 1)^2 y_2 + 2x(x^2 + 1)y_1 = 2$
CBSE Solutions for Class 12 Mathematics Continuity and Differentiability
- Continuity and Differentiability Class 12 EXERCISE 5.1
- Continuity and Differentiability Class 12 EXERCISE 5.2
- Continuity and Differentiability Class 12 EXERCISE 5.3
- Continuity and Differentiability Class 12 EXERCISE 5.4
- Continuity and Differentiability Class 12 EXERCISE 5.5
- Continuity and Differentiability Class 12 EXERCISE 5.6
Chapters in CBSE - Class 12 Mathematics
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