A general solution of ≤ft(dy/dx)=2x+3y, y(0)=0 is

Correct Answer is : 2e−3y+3e2x=52e^{-3y}+3e²x}=52e−3y+3e2x=5

Correct Answer is : 2e−3y+3e2x=52e^{-3y}+3e²x}=52e−3y+3e2x=5

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UPSC NDA Math Model Paper 3

Question : 43 of 120

Marks: +1, -0

\log\left(\frac{dy}{dx}\right)=2x+3y,\quad y(0)=0

Solution:

Given differential equation is

\log\left(\frac{dy}{dx}\right)=2x+3y

\Rightarrow\frac{dy}{dx}=e^{2x+3y}

\Rightarrow\frac{dy}{dx}=e^{2x}e^{3y}

\Rightarrow\frac{dy}{e^{3y}}=e^{2x}dx

Integrating both sides, we get

\Rightarrow\int e^{-3y}dy=\int e^{2x}dx

\Rightarrow\frac{e^{-3y}}{-3}=\frac{e^{2x}}{2}+c

\Rightarrow2e^{-3y}=-3e^{2x}-6c

.......(1)

By using y = 0 when x = 0, we get

⇒ 2 = -3 – 6c

∴ c = -5/6

Put the value of

c

in equation

1^{\text{st }}

\Rightarrow2e^{-3y}=-3e^{2x}-6\left(-\frac{5}{6}\right)

\Rightarrow2e^{-3y}=-3e^{2x}+5

\Rightarrow2e^{-3y}+3e^{2x}=5

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