Problem 12
Question
Solve the differential equations. $$\frac{d y}{d x}=3 x^{2} e^{-y}$$
Step-by-Step Solution
Verified Answer
\( y = \ln(x^3 + C) \) is the solution.
1Step 1: Separate Variables
To solve the differential equation \( \frac{d y}{d x} = 3x^2 e^{-y} \), we first need to separate the variables. We rewrite the equation to isolate terms involving \(y\) on one side and terms involving \(x\) on the other side. This gives us: \[ e^y rac{d y}{d x} = 3x^2 \] Now, we can rearrange to get: \[ e^y \, dy = 3x^2 \, dx \]
2Step 2: Integrate Both Sides
Now that we have separated the variables, we can integrate both sides of the equation. The left side with respect to \(y\) and the right side with respect to \(x\): \[ \int e^y \, dy = \int 3x^2 \, dx \] Integrating both sides gives: \[ e^y = x^3 + C \] where \(C\) is the constant of integration.
3Step 3: Solve for y
To express \(y\) explicitly, we take the natural logarithm of both sides to solve for \(y\), giving: \[ y = \ln(x^3 + C) \] This represents the general solution to the differential equation, expressing \(y\) in terms of \(x\).
Key Concepts
Separation of VariablesIntegrationConstant of IntegrationGeneral Solution
Separation of Variables
When you encounter a differential equation, one of the first techniques you might use is the separation of variables. This method involves rearranging the equation so that each variable appears on a different side. For example, in the exercise \(\frac{dy}{dx} = 3x^2 e^{-y}\), our goal is to have all terms involving \(y\) on one side and all terms involving \(x\) on the other.
This is achieved by multiplying or dividing both sides accordingly. By isolating the terms, the problem becomes easier to tackle through integration. The separation of variables is not only a technique but also a creative way to make a complex equation more manageable. Recognizing when and how to apply it will be crucial as you delve deeper into solving differential equations.
This is achieved by multiplying or dividing both sides accordingly. By isolating the terms, the problem becomes easier to tackle through integration. The separation of variables is not only a technique but also a creative way to make a complex equation more manageable. Recognizing when and how to apply it will be crucial as you delve deeper into solving differential equations.
Integration
Once the equation is separated, our next step is to integrate both sides. Integration is the mathematical process of finding the antiderivative. For each side of the equation, integrate the term with respect to its respective variable.
In the example, we integrate \( e^y \, dy \) and \( 3x^2 \, dx \). These integrations yield the solutions \( e^y + C_1 \) and \( x^3 + C_2 \) respectively, where \( C_1 \) and \( C_2 \) are constants of integration. For simplicity, we often combine them into a single constant \( C \).
In the example, we integrate \( e^y \, dy \) and \( 3x^2 \, dx \). These integrations yield the solutions \( e^y + C_1 \) and \( x^3 + C_2 \) respectively, where \( C_1 \) and \( C_2 \) are constants of integration. For simplicity, we often combine them into a single constant \( C \).
- Use known antiderivative formulas: This helps speed up the calculation process.
- Pay attention to the differential: Ensure that you integrate with respect to the correct variable.
Constant of Integration
When you integrate, a constant of integration \( C \) appears. This constant represents any fixed number that could have been added to a function before differentiation. Since derivative formulas eliminate constants, integrating brings back this unknown value.
In our example, after integrating, the equation becomes \( e^y = x^3 + C \). This \( C \) accounts for the infinite number of curves that can satisfy the differential equation. Without it, we would have just one particular solution. By including \( C \), you acknowledge all possible solutions and later adjust it based on specific initial conditions to find a specific solution. This concept is essential because it highlights the presence of multiple valid solutions in calculus.
In our example, after integrating, the equation becomes \( e^y = x^3 + C \). This \( C \) accounts for the infinite number of curves that can satisfy the differential equation. Without it, we would have just one particular solution. By including \( C \), you acknowledge all possible solutions and later adjust it based on specific initial conditions to find a specific solution. This concept is essential because it highlights the presence of multiple valid solutions in calculus.
General Solution
The resulting expression after completing the steps of separation and integration is known as the general solution. It contains the constant of integration, representing a family of curves. In our example, the general solution is \( y = \ln(x^3 + C) \).
This solution is general because it can adapt to particular situations by determining the value of \( C \) when given an initial condition or boundary condition, such as \( y(a) = b \).
This solution is general because it can adapt to particular situations by determining the value of \( C \) when given an initial condition or boundary condition, such as \( y(a) = b \).
- General solutions provide a broad understanding of how the solution behaves.
- Allows adaptability to specific scenarios by setting suitable conditions.
Other exercises in this chapter
Problem 11
Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=\left(x^{2}-2 x+2\right) e^{x}$$
View solution Problem 12
When is a polynomial \(f(x)\) of smaller order than a polynomial \(g(x)\) as \(x \rightarrow \infty ?\) Give reasons for your answer.
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Use the definitions of cosh \(x\) and \(\sinh x\) to show that $$\cosh ^{2} x-\sinh ^{2} x=1$$
View solution Problem 12
Find the values. $$\cot \left(\sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right)$$
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