Question

Assertion :Statement- 1 : The differential equation $x\frac{dy}{dx}+(x^2-5)y=e^x$ is a linear equation. Reason: Statement- 2 : Every first order differential equation is a linear equation

• A. Statement -1 is True, Statement -2 is True ; Statement -2 is a correct explanation for Statement -1
• B. Statement-1 is True, Statement-2 is True ; Statement-2 is NOT a correct explanation for Statement-1
• C. Statement -1 is True, Statement -2 is False
• D. Statement -1 is False, Statement -2 is True

Answer 1

The correct option is C Statement -1 is True, Statement -2 is False

Given, Differential equation as $x\frac{dy}{dx}+(x^2-5)y=e^x$
$\Rightarrow \frac{dy}{dx}+\frac{(x^2-5)}{x}y=\frac{e^x}{x}$

For equation to be linear,
a)The powers of dependent variable and its derivative to be power of one.
b)The coefficient of dependent variable should be either function of independent variable or constant.

Since in our equation the power of dependent variable and its derivative is one.

The coefficients of the dependent variable is also function of x only.so, the given equation is linear.

Reason is false,since it is not necessary for every first order to be linear equation.
Hence, Assertion is true and Reason is false.