AP EAMCET · Maths · Differential Equations
The general solution of the differential equation \(\frac{d y}{d x}+x y=4 x-2 y+8\) is
- A \(y=4-c e^{-\frac{(x+2)^2}{2}}\)
- B \(y=8+c e^{\frac{-x^2}{2}-2 x}\)
- C \(y=c e^{-(x+2)^2}+x\)
- D \(y+2 x=c e^{-\frac{x}{2}-2 x}\)
Answer & Solution
Correct Answer
(A) \(y=4-c e^{-\frac{(x+2)^2}{2}}\)
Step-by-step Solution
Detailed explanation
\(\frac{d y}{d x}+x y=4 x-2 y+8\) \(\frac{d y}{d x}+(x+2) y=4(x+2)\) \(IF = e^{\int (x+2) dx} = e^{\frac{x^2}{2}+2x}\) \(y \cdot e^{\frac{x^2}{2}+2x} = \int 4(x+2) e^{\frac{x^2}{2}+2x} dx + C\) \(y \cdot e^{\frac{x^2}{2}+2x} = 4 e^{\frac{x^2}{2}+2x} + C\)…
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