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3. Find the general solution of the following differential equation. 1 (1 += sin²z) de +(+²2) dy-0 PROBLEM STEP BY STEP 2y EXPLAIN IN WORDS HOW TO SOLVE THIS 3. Find the general solution of the following differential equation . 1 ( 1 + = sin²z ) de + ( + ²2 ) dy - 0 PROBLEM STEP BY STEP 2y EXPLAIN IN WORDS HOW TO SOLVE THIS​

3. Find the general solution of the following differential equation. 1 (1 += sin²z-example-1
User AmitSri
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1 Answer

19 votes
19 votes

Multiply both sides by
y to get an exact equation.


\left(1 + \frac{x^3}y \sin^2(x)\right) \, dx + \frac1y \left(x + \frac1{\cos^2(2y)}\right) \, dy = 0


\implies \underbrace{(y + x^3 \sin^2(x))}_(M(x,y)) \, dx + \underbrace{(x + \sec^2(2y))}_(N(x,y)) \, dy = 0

This ODE is exact since
(\partial M)/(\partial y) = (\partial N)/(\partial x). Then the solution is given by an implicit function


f(x,y) = C

Taking differentials on both sides by the chain rule gives


(\partial f)/(\partial x) \, dx + (\partial f)/(\partial y) \, dy = 0

so that we have the system of partial differential equations


(\partial f)/(\partial x) = M = y + x^3 \sin^2(x)


(\partial f)/(\partial y) = N = x + \sec^2(2y)

Integrate both sides of the first of these equations with respect to
x to recover
f.


\displaystyle \int (\partial f)/(\partial x) \, dx = \int (y + x^3 \sin^2(x)) \, dx


\implies f(x,y) = xy + g(x) + h(y)

where
g(x) is the antiderivative of
x^3\sin^2(x) (and is easy enough to compute by parts).

Differentiating both sides with respect to
y gives


(\partial f)/(\partial y) = x + (dh)/(dy) = x + \sec^2(2y)


\implies (dh)/(dy) = \sec^2(2y)


\implies h(y) = \frac12 \tan(2y) + C

Then the general solution to the ODE is


f(x,y) = \boxed{xy + g(x) + \frac12 \tan(2y) = C}

User Collusionbdbh
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3.2k points