Differential Equations
Differential Equations
Allen Star Batch
Grade 12
Question:
If $y_1, y_2$ are the solution of the differential equation $\frac{dy}{dx} + P(x)y = Q(x)$, then:
$y = y_1 + c(y_1 - y_2)$ is the general solution of equation
$y = y_1 + c(y_1 + y_2)$ is the general solution of equation
$\alpha y_1 + \beta y_2$ is a solution of $\alpha + \beta = 1$
$\alpha y_1 + \beta y_2$ is a solution of $\alpha - \beta = 1$
Step-by-Step Solution
Key Concept: The solution space of a linear ODE is a vector space; any linear combination of solutions with coefficients summing to 1 is also a solution.
The linear system has equations $\frac{dy}{dx} + Py = 0$, $\frac{dy_1}{dx} + Py_1 = 0$, and $\frac{dy_2}{dx} + Py_2 = 0$. Taking differences and noting that $\alpha y_1 + \beta y_2$ is a solution with $\alpha + \beta = 1$ gives the general form $y = y_1 + c(y_1 - y_2)$ where $c$ is arbitrary.
Correct Answer: 1,3