Question

Solution of \[ p^3y+2px-y=0 \] is \(____\), where \[ p=\frac{dy}{dx}. \]

• A. \(y^2=2cx+c^3\)
• B. \(y=cx+c^2\)
• C. \(x=cy^2+c^3\)
• D. \(x=2cy+c^3\)

Hint:

For differential equations involving \(p=\frac{dy}{dx}\), verify the given option by differentiating it and eliminating the arbitrary constant.

Answer 1

The Correct Option is A

Concept:
Given differential equation is: \[ p^3y+2px-y=0 \] where \[ p=\frac{dy}{dx} \] Here, the equation contains \(x\), \(y\), and \(p\). We need to find the solution by checking which option satisfies the given differential equation.

Step 1: Take option (A).
Given: \[ y^2=2cx+c^3 \] Here \(c\) is an arbitrary constant.

Step 2: Differentiate with respect to \(x\).
Differentiating both sides: \[ \frac{d}{dx}(y^2)=\frac{d}{dx}(2cx+c^3) \] Since \(c\) is constant: \[ 2y\frac{dy}{dx}=2c \] Using: \[ p=\frac{dy}{dx} \] we get: \[ 2yp=2c \] \[ yp=c \] Therefore: \[ c=yp \]

Step 3: Substitute \(c=yp\) in the given solution.
Original option is: \[ y^2=2cx+c^3 \] Substitute: \[ c=yp \] \[ y^2=2(yp)x+(yp)^3 \] \[ y^2=2pxy+y^3p^3 \]

Step 4: Rearrange the equation.
Bring all terms to one side: \[ y^2-2pxy-y^3p^3=0 \] Taking \(y\) common: \[ y(y-2px-y^2p^3)=0 \] For \(y\neq0\), we get: \[ y-2px-y^2p^3=0 \] Multiplying by \(-1\): \[ y^2p^3+2px-y=0 \] This is the same as the given differential equation: \[ p^3y+2px-y=0 \] Hence, option (A) satisfies the given differential equation.

Step 5: Final answer.
Therefore, the solution of the given differential equation is: \[ y^2=2cx+c^3 \] \[ \therefore \text{Correct Answer is (A)} \]