Question

Solve the following ordinary differential equations by separating variables and integration.
1. y'sinx=ylny, the boundary condition (b.c.) is that the function passes through (y=e,x=pi/3)
2. y'+2xy2=0, b.c. (y=1;x=2)
3. y'-xy=x, b.c. (y=1;x=0)

Answer 1

1) Here, we have

separating variables,

integrating on both sides,

Let us make the substitution,

Then,

Substituting this back in our equation:

ie,

Where C is the constant of integration

Or it could be simply written as:

The C has culminated into A during the above operation.

Applying B.C,

ie,

Hence the complete solution is:

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2)

Separating the variables,

integrating,

ie,

Using give boundary condition:

ie,

Hence, the complete solution is:

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3)

Separating the variables:

Integrating:

Taking exponential on both sides:

Where A is the constant we have figure out using the Boundary condition,

ie,

Therefore, the complete solution is: