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Magazine Chapter 3: Problem 8
In Problems, solve the given differential equation by using the substitution \(u=y^{\prime}\). $$ y^{2} y^{\prime \prime}=y^{\prime} $$
Short Answer
Expert verified
Use substitution \( u = y' \) to form a separable equation, solve for \( u = y' \), and integrate to find \( y \).
Step by step solution
01
Substitute u for y'
We start by using the substitution \( u = y' \). This means that \( y'' = u' = \frac{du}{dx} \). This substitution turns the original differential equation \( y^2 y'' = y' \) into \( y^2 u' = u \).
02
Rearrange in terms of u and y
Reformulate the equation \( y^2 u' = u \) into \( u' = \frac{u}{y^2} \). This step isolates \( u' \) on one side of the equation.
03
Separate the variables
Recognizing this as a separable differential equation, separate the variables by rewriting it as \( u' = \frac{du}{dx} = \frac{u}{y^2} \), which rearranges to \( \frac{du}{u} = \frac{dx}{y^2} \).
04
Integrate both sides
Integrate both sides of the equation \( \int \frac{du}{u} = \int \frac{dx}{y^2} \). This gives \( \ln|u| = \frac{C}{y} + C_1 \) where \( C_1 \) is a constant of integration.
05
Solve for u
Exponentiate both sides to solve for \( u \). This yields \( u = e^{C_1} e^{\frac{C}{y}} \). Let \( K = e^{C_1} \), simplifying to \( u = K e^{\frac{C}{y}} \).
06
Substitute back to find y'
Recall \( u = y' \), replace \( u \) with \( y' \) to get \( y' = K e^{\frac{C}{y}} \).
07
Integrate to find y
Integrate \( y' = K e^{\frac{C}{y}} \) with respect to \( y \) to find \( y \) as a function of \( x \). The integration might require special techniques depending on the form of the resulting integral.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method is a handy tool in solving differential equations, especially when dealing with complex derivatives. Here, we start by identifying a part of the equation to replace with a new variable, simplifying the mathematics involved. In our problem, we have substituted \( u = y' \). This simple replacement transforms the complicated second-derivative problem into one involving only first derivatives.
This greatly simplifies our computations.
This greatly simplifies our computations.
- Identify the part of the equation to substitute (e.g., \( y' \))
- Reformulate the equation using the substitution
- Solve the simpler equation
Separable Differential Equations
Separable differential equations allow us to split variables algebraically. This is achievable with equations that, once simplified, let us group all \( x \)-related terms and \( y \)-related terms separately. In our exercise, after substituting \( u = y' \), the equation becomes separable to \( \frac{du}{u} = \frac{dx}{y^2} \).
Here's how you separate them:
Here's how you separate them:
- Rearrange to isolate terms: both sides should hold variables and their differentials
- Invert and solve to have \( dx \) terms on one side and \( du \) terms on another
Constant of Integration
When integrating, remember each integral can have a constant of integration because integration is the inverse of differentiation. In the step where you integrate \( \int \frac{du}{u} = \int \frac{dx}{y^2} \), a constant, \( C_1 \), emerges in the solution: \( \ln|u| = \frac{C}{y} + C_1 \). This constant accounts for any number that differentiates to zero and must be considered for the full spectrum of solutions.
Some key points to remember include:
Some key points to remember include:
- The constant can adjust vertically throughout solutions
- It represents the family of solutions instead of a single solution
- Sometimes expressed in terms of \( x \) or replaced with a convenient transformation
Integration Techniques
Integrating functions requires a selection from various techniques: substitution, partial fractions, and trigonometric identities, to name a few. In our exercise, integrating \( y' = K e^{\frac{C}{y}} \) might involve substitution, particularly if the integral appears complex. Typically:
- Examine for standard derivative inverses
- Use techniques that simplify the function into recognizable patterns
- Let \( y \) or another variable be \( = \int \, \) to clearly isolate \( x \)
