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Please solve RD Sharma Class 12 Chapter 21 Differential Equation Exercise 21.10 Question 6 maths textbook solution.

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Answer : y=(2 x-1)+C e^{-2 x}

Hint : To solve this equation we will use differentiate different.

Give: \frac{dy}{dx}+2y=4x

Solution : \frac{dy}{dx}+Py=Q

               \begin{aligned} &P=2, Q=4 x \\ &I f=e^{\int P d x} \\ &=e^{\int 2 d x} \\ &=e^{-2 x} \end{aligned}

               \begin{aligned} &y \times I f=\int Q \times I f d x+C \\ &y e^{2 x}=\int 4 x e^{2 x} d x+C \\ &y e^{2 x}=4 \int x e^{2 x} d x+C \end{aligned}

                \begin{aligned} &y e^{2 x}=4\left[\frac{x e^{2 x}}{2}\right]-\left[\frac{e^{2 x}}{2} d x\right]+C \\ &y e^{2 x}=2 x e^{2}-2 \frac{e^{2 x}}{2}+C \\ &y=(2 x-1)+C e^{-2 x} \end{aligned}

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