The solution of ≤ft(d y/d x)=2 x-5 y, y(0)=0 is

Correct Answer is : 5e2x−2e5y=3 5 e² x}-2 e^{5 y}=35e2x−2e5y=3

Correct Answer is : 5e2x−2e5y=3 5 e² x}-2 e^{5 y}=35e2x−2e5y=3

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MHTCET 5 May 2025 Shift 2 Paper

Question : 133 of 150

Marks: +1, -0

\log \left(\frac{d y}{d x}\right)=2 x-5 y, y(0)=0

Solution:

Concept:

Separate the variables and use the initial condition to find the particular solution.

Explanation:

Given:

\log \left(\frac{dy}{dx}\right)=2x-5y

Taking antilog:

\frac{dy}{dx}=e^{2x-5y}=e^{2x}\cdot e^{-5y}

Separating variables:

e^{5y}\,dy=e^{2x}\,dx

Integrating both sides:

\frac{e^{5y}}{5}=\frac{e^{2x}}{2}+C

Using

y(0)=0

, i.e.

x=0

y=0

\frac{1}{5}=\frac{1}{2}+C

C=\frac{1}{5}-\frac{1}{2}=-\frac{3}{10}

Substitute

C

\frac{e^{5y}}{5}=\frac{e^{2x}}{2}-\frac{3}{10}

Multiply by

10

2e^{5y}=5e^{2x}-3

Rearranging:

5e^{2x}-2e^{5y}=3

Answer:

Option B:

5e^{2x}-2e^{5y}=3

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