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Magazine Chapter 31: Problem 7
solve the given differential equations. $$t^{3} d r+r t^{2} d t=t d r-r d t$$
Short Answer
Expert verified
The solution is \(r = \frac{K}{|t|^{1/2} |t-1| |t+1|^{1/2}}\).
Step by step solution
01
Equation Setup and Separation
First, rewrite the given differential equation as: $$t^{3} d r + r t^{2} d t = t d r - r d t.$$ Our objective is to group terms involving the same differentials. Isolate all terms involving \(dr\) and \(dt\) respectively, which gives us: $$ (t^3 - t) dr = (-r dt + r t^2 dt).$$
02
Simplifying and Factoring
Factor both sides with respect to the derivatives: \(dr\) and \(dt\). We then have: \((t^3 - t) dr = -r(t^2 + 1) dt\). Now, divide both sides by \((t^3 - t)(t^2 + 1)\) to separate the variables: $$\frac{dr}{r} = -\frac{(t^2 + 1)}{(t^3 - t)} dt.$$
03
Integrate Both Sides
Integrate both sides of the equation with respect to their own variables. This yields: $$\int \frac{1}{r} dr = -\int \frac{t^2 + 1}{t^3 - t} dt.$$ For the left side, the integral is \(\ln |r| + C_1\). The right side requires partial fraction decomposition for integration.
04
Partial Fraction Decomposition and Further Integration
To solve \( \int \frac{t^2 + 1}{t^3 - t} dt \), first factor the denominator as \(t(t^2 - 1) = t(t-1)(t+1)\). Perform partial fraction decomposition: \( \frac{t^2 + 1}{t(t-1)(t+1)} = \frac{A}{t} + \frac{B}{t-1} + \frac{C}{t+1} \). Solve to find constants \(A, B,\) and \(C\).
05
Solving Partial Fractions
Equate the numerators after cross-multiplying: \(t^2 + 1 = A(t-1)(t+1) + Bt(t+1) + Ct(t-1)\). Substitute suitable values for \(t\) (such as \(t=0, 1, -1\)) to find \(A, B, \) and \(C\). Calculating gives \(A = 1/2, B = 1, C = 1/2\).
06
Integration of Partial Fractions
Integrate each fraction separately: $$\int \frac{A}{t} dt = \frac{1}{2} \ln |t|,$$ $$\int \frac{B}{t-1} dt = \ln |t-1|,$$ and $$\int \frac{C}{t+1} dt = \frac{1}{2} \ln |t+1|.$$ Combine these integrals to get \(-\frac{1}{2} \ln |t| - \ln |t-1| - \frac{1}{2} \ln |t+1| + C_2\).
07
Combine Results
Combine the integral results: \(\ln |r| + C_1 = - \frac{1}{2} \ln |t| - \ln |t-1| - \frac{1}{2} \ln |t+1| + C_2\). Simplifying and combining the constants yields the solution as \(r = \frac{K}{|t|^{1/2} |t-1| |t+1|^{1/2}}\), where \(K\) is a constant obtained from \(C_1\) and \(C_2\).
08
Final Answer
Write the final expression of the answer which encompasses the integration constant: \(r = \frac{K}{|t|^{1/2} |t-1| |t+1|^{1/2}}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Separation of Variables
Separation of variables is a technique used to solve differential equations. The goal is to rewrite the equation in a form where each variable and its differential are on separate sides of the equation. This allows easier integration.
The process involves three main steps:
The process involves three main steps:
- Identify and rearrange the terms: Start by isolating the terms involving derivatives on each side of the equation.
- Divide through by a common factor: Often, the differential equation can be modified by dividing each term by a factor that separates the variables.
- Integrate: Once the variables are separated, integrate each side with respect to its variable.
Partial Fraction Decomposition
Partial fraction decomposition is a method used to break down complex rational expressions into simpler fractions. This simplification is particularly useful when integrating rational functions.
The main process includes:
The main process includes:
- Factor the denominator: Begin by factoring the polynomial in the denominator completely.
- Set up the decomposition: Express the function as a sum of fractions, each with a simpler denominator.
- Solve for constants: Determine the coefficients of these fractions by making the numerators equal.
Integration Techniques
Integration is a fundamental technique in calculus, crucial in solving differential equations. Here, both basic integration and more advanced techniques are used.
Key techniques include:
Key techniques include:
- Basic Integration: Direct integration of functions like \(\int \frac{1}{r} dr = \ln |r|\).
- Logarithmic Integration: Used in expressions like \(\int \frac{1}{t} dt = \ln |t|\).
- Integrating decomposed fractions: Once decomposed, each fraction can be integrated individually, making the process simpler.
