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Magazine Chapter 14: Problem 28
Find the solution of $$ (5 x+y-7) \frac{d y}{d x}=3(x+y+1) $$.
Short Answer
Expert verified
y = C(x + y - 7)^3 - x - 1
Step by step solution
01
- Separate Variables
Rewrite the equation to separate the variables. Divide both sides by (5x + y - 7): \[\frac{d y}{d x} = \frac{3(x + y + 1)}{5x + y - 7}\] We can move terms involving y on one side and terms involving x on the other side.
02
- Rearrange the Equation
Rearrange the equation as follows: \[\frac{d y}{x + y + 1} = \frac{3d x}{5x + y - 7}\]
03
- Integrate Both Sides
Integrate both sides of the equation. Recall that you may need to use substitution methods. For example, let z = x + y + 1. Thus, dz = dx + dy: \ \[\frac{dy}{z} = \frac{3dx}{z - 8}\] Next, separate and integrate both parts.
04
- Apply Integration
Integrate both sides: \[\frac{1}{z}dz = 3 \frac{1}{(z-8)}dx\] After integrating: \[\text{ln}|z| = 3\text{ln}|z-8| + C\]
05
- Solve for y
Back substitute z = x + y + 1: \[\text{ln}|x + y + 1| = 3\text{ln}|(x + y - 7)| + C\] Exponentiate both sides to solve for y.
06
- Simplify the Equation
Combine constants into a single constant: \[x + y + 1 = K(x + y - 7)^3\] where K is a constant.
07
- Determine Solution
Solve for y: \[y = C_1 (x + y - 7)^3 - x - 1\] where \(C_1\) is a new constant derived from K.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Variable Separation
Variable separation is a strategy used to solve differential equations by isolating variables on different sides of the equation. In the exercise given, we rewrote the equation to isolate terms involving y on one side and terms involving x on the other.
- Start by rewriting the equation.
- Then, divide both sides to move y terms on one side and x terms on the other.
Integration Methods
Integration methods involve various techniques to find the integral of functions. In our example, after separating variables, we move on to the integration stage.
- We integrated both sides of the equation after transforming it into a convenient form.
- This often involves direct integration or special techniques like substitution.
Substitutions
Substitution is a powerful tool in solving differential equations and integrals. In our specific problem, we set z = x + y + 1 to make the differential equation easier to handle.
- Substitution often involves introducing a new variable to simplify complications.
- It converts a difficult problem into a simpler one that is easier to solve.
Differential Equation Solutions
The solution of a differential equation is a function that satisfies the equation. After substituting and integrating, we resolved our differential equation solution by back-substituting the original variables.
The general solution gives a family of functions dependent on a constant, dictated by initial conditions or additional constraints. This problem鈥檚 final form was y expressed in relation to x after completing all steps.
- Upon finding an integrated form, substitute back to the original variables if needed.
- Simplify the equation to find a general or specific solution.
The general solution gives a family of functions dependent on a constant, dictated by initial conditions or additional constraints. This problem鈥檚 final form was y expressed in relation to x after completing all steps.
