Solve the differential equation.
$\left(\frac{1}{7} \sec x\right) \frac{d y}{d x}=2 e^{y+3 \sin x}$
The solution is given by the equation $\square$ .
(Type an equation. Type an exact answer.)
Answer (2)
To solve the given differential equation, we isolate the variables and integrate both sides. This process leads to the solution presented as y = − ln ( C − 3 14 e 3 s i n x ) .
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Separate variables in the given differential equation.
Integrate both sides of the separated equation.
Solve for y in terms of x .
The solution to the differential equation is: y = − ln ( C − 3 14 e 3 s i n x )
Explanation
Problem Setup We are given the differential equation ( 7 1 sec x ) d x d y = 2 e y + 3 s i n x . Our goal is to solve for y as a function of x .
Isolating dy/dx First, we rewrite the equation to isolate d x d y :
Rewritten Equation d x d y = 14 e y + 3 s i n x cos x
Separating Variables Next, we separate the variables. We can rewrite the exponential term as e y + 3 s i n x = e y e 3 s i n x . Then, we have:
Rewriting Exponential Term d x d y = 14 e y e 3 s i n x cos x
Dividing by e^y Now, divide both sides by e y to get:
Separated Variables e − y d y = 14 e 3 s i n x cos x d x
Integrating Both Sides Now, we integrate both sides:
Integral Equation ∫ e − y d y = ∫ 14 e 3 s i n x cos x d x
Evaluating Integrals The integral on the left side is straightforward: ∫ e − y d y = − e − y + C 1 . For the integral on the right side, we use a substitution. Let u = 3 sin x , so d u = 3 cos x d x . Then, the integral becomes:
Substitution and Integration ∫ 14 e 3 s i n x cos x d x = ∫ 3 14 e u d u = 3 14 e u + C 2 = 3 14 e 3 s i n x + C 2
Combining Results So, we have:
Integrated Equation − e − y = 3 14 e 3 s i n x + C
Multiplying by -1 Multiply both sides by − 1 :
Simplified Equation e − y = − 3 14 e 3 s i n x − C
Rewriting Constant We can rewrite − C as another constant, say C ′ :
Equation with New Constant e − y = C ′ − 3 14 e 3 s i n x
Taking Natural Logarithm Take the natural logarithm of both sides:
Logarithmic Equation − y = ln ( C ′ − 3 14 e 3 s i n x )
Solving for y Multiply by − 1 to solve for y :
Final Solution y = − ln ( C ′ − 3 14 e 3 s i n x )
Replacing Constant We can replace C ′ with C to represent the constant of integration:
Final Answer y = − ln ( C − 3 14 e 3 s i n x )
Examples
Differential equations are used in physics to describe the motion of objects, such as a pendulum or a falling body. They are also used in engineering to design circuits and control systems. In biology, they can model population growth and the spread of diseases. Solving differential equations allows us to understand and predict the behavior of these systems over time.
