Question

Given are two ordinary differential equations
P: $\dfrac{dy}{dx} + x = x \sin y$
Q: $\dfrac{dy}{dx} + x y = e^{x} y$
The correct choice is:

• A. P is linear; Q is nonlinear
• B. P is nonlinear; Q is linear
• C. Both P and Q are linear
• D. Both P and Q are nonlinear

Hint:

To test linearity, check if $y$ and its derivatives appear only in the first degree and are not inside trigonometric, exponential, or product terms like $yy'$ or $y^2$.

Answer 1

The Correct Option is D

Step 1: Recall the definition.
A differential equation is linear if the dependent variable ($y$) and its derivatives appear only in the first power and are not multiplied/divided by each other or in nonlinear functions like $\sin y$, $e^y$, etc.
Step 2: Analyze equation P.
P: $\dfrac{dy}{dx} + x = x \sin y$ Here, the term $\sin y$ makes the equation nonlinear (since $y$ appears inside a trigonometric function). Hence, P is nonlinear.
Step 3: Analyze equation Q.
Q: $\dfrac{dy}{dx} + x y = e^x y$ This can be rearranged as: \[ \dfrac{dy}{dx} = y(e^x - x) \] Here, $y$ is multiplied by a function of $x$, which still keeps it linear because $y$ appears only to the first power. Wait carefully — but note: in the term $x y$, $y$ is simply multiplied by $x$ (independent variable), which is fine. However, the right-hand side is $e^x y$, still only linear in $y$. Correction: Q is actually linear. So final: P nonlinear, Q linear. Final Answer: \[ \boxed{\text{(B) P is nonlinear; Q is linear}} \]