Method of DifferentiationmediumFree

Method of Differentiation: Satisfies Axy Equal

JEE Maths question with a full step-by-step solution.

Question
If y=ex+e−xy = e^{\sqrt{x}}+e^{-\sqrt{x}} satisfies Axy′′+By′+Cy=0Axy''+By'+Cy = 0, then A+B+CA+B+C is equal to.
Solution
Answer: 5
Step 1: First derivative
y′=ex−e−x2x  ⟹  2x y′=ex−e−xy' = \frac{e^{\sqrt{x}}-e^{-\sqrt{x}}}{2\sqrt{x}} \implies 2\sqrt{x}\,y' = e^{\sqrt{x}}-e^{-\sqrt{x}}
 Step 2: Differentiate again Differentiating 2x y′=ex−e−x2\sqrt{x}\,y' = e^{\sqrt{x}}-e^{-\sqrt{x}} with respect to xx:
y′x+2x y′′=ex+e−x2x=y2x\frac{y'}{\sqrt{x}}+2\sqrt{x}\,y'' = \frac{e^{\sqrt{x}}+e^{-\sqrt{x}}}{2\sqrt{x}} = \frac{y}{2\sqrt{x}}
 Step 3: Form the differential equation Multiplying by 2x2\sqrt{x}:
2y′+4xy′′=y  ⟹  4xy′′+2y′−y=02y'+4xy'' = y \implies 4xy''+2y'-y = 0
 So A=4A = 4, B=2B = 2, C=−1C = -1, and A+B+C=5A+B+C = 5. Answer: 5
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