Differential Equations
Differential Equation
nta_abhyas_2025
Grade 12
Question:
The order of the differential equation whose general solution is given by $y = (c_1 + c_2x)\cos(x + c_3) - c_4e^{-7x}$, where $c_1, c_2, c_3, c_4$ & $c_5$ are arbitrary constants, is
Step-by-Step Solution
Key Concept: Integrating both sides of a differential equation and applying boundary conditions to find constants of integration
Integrating both sides of $\frac{f(0)}{x} = x + C$ with the condition $f(1) = \frac{1}{2}$, we first establish that $\frac{1}{x} = 1 + C$, giving $C = -\frac{1}{x}$. From $f(0) = x^2 - \frac{1}{x}$ and using the boundary condition $f(2) = 4 - \frac{1}{4} = \frac{15}{4}$, we verify the integration constant. Therefore $3f(2) = 3 \times \frac{15}{4} = \frac{45}{4}$, which simplifies to 8 when properly evaluated.
Correct Answer: 3