Problem 14
Question
Sketch a representative family of solutions for each of the following differential equations. (a) \(\frac{d y}{d t}=t^{2}\) (b) \(\frac{d y}{d t}=y^{2}\)
Step-by-Step Solution
Verified Answer
The solution for equation (a) is \(y(t)=\frac{1}{3}t^3+C\), forming a family of cubic curves. The solution for equation (b) is \(y(t)=-\frac{1}{t + C}\), forming a family of hyperbola.
1Step 1: Solving Equation (a)
To solve \(\frac{d y}{d t}=t^{2}\), integrate both sides of the equation with respect to \(t\), this gives the solution \(y(t)=\int t^2 dt = \frac{1}{3}t^3+C\), where \(C\) is the constant of integration. These represent a family of cubic curves in the \(y-t\) plane.
2Step 2: Sketching Equation (a) Solution
Sketch the family of solution curves \(y(t)=\frac{1}{3}t^3+C\). As \(C\) varies, the entire set forms a family of cubic curves. All curves will share the common property that the slope of the curve at any point \((x,y)\) on it is given by \(t^2\).
3Step 3: Solving Equation (b)
To solve the equation \(\frac{d y}{d t}=y^{2}\), separate the variables: \(\frac{1}{y^2} dy = dt\). Integrating both sides then gives: \(-\frac{1}{y} = t + C\), where \(C\) again is the constant of integration. This can be rewritten as \(y(t) = -\frac{1}{t+C}\). This represents a family of hyperbola in the \(y-t\) plane.
4Step 4: Sketching Equation (b) Solution
Sketch the family of solution curves \(y(t) = -\frac{1}{t+C}\). As \(C\) varies, the entire set forms a family of hyperbola, and all curves share the property that the slope of the tangent line at a point \((t, y)\) is given by \(y^2\).
Key Concepts
IntegrationFamily of SolutionsSeparation of Variables
Integration
Integration is a fundamental concept in calculus, widely used to solve differential equations.Integrating involves finding a function whose derivative is the given function.This process is called anti-differentiation.
To solve the differential equation \( \frac{d y}{d t}=t^{2} \), we integrate both sides with respect to \( t \).This means calculating the integral of \( t^2 \) with respect to \( t \), which gives the solution \( y(t)=\int t^2 dt = \frac{1}{3}t^3+C \).Here, \( C \) is known as the constant of integration, representing an infinite set of solutions or a 'family' of functions.
To solve the differential equation \( \frac{d y}{d t}=t^{2} \), we integrate both sides with respect to \( t \).This means calculating the integral of \( t^2 \) with respect to \( t \), which gives the solution \( y(t)=\int t^2 dt = \frac{1}{3}t^3+C \).Here, \( C \) is known as the constant of integration, representing an infinite set of solutions or a 'family' of functions.
- Integration helps in reversing the effect of differentiation.
- It allows us to find specific functions that satisfy certain initial or boundary conditions.
- The constant of integration \( C \) ensures that all possible solutions to a differential equation are captured.
Family of Solutions
A family of solutions to a differential equation includes all possible solutions characterized by different values of the constant of integration \( C \).In the case of the first differential equation, the family of solutions forms cubic curves \( y(t)=\frac{1}{3}t^3 + C \).
The constant \( C \) can take any real value, causing the curves to shift vertically in the \( y-t \) plane.Similarly, for the second equation \( y(t) = -\frac{1}{t+C} \), the family of solutions are hyperbolas, where each curve in the family is defined by a different \( C \).
The constant \( C \) can take any real value, causing the curves to shift vertically in the \( y-t \) plane.Similarly, for the second equation \( y(t) = -\frac{1}{t+C} \), the family of solutions are hyperbolas, where each curve in the family is defined by a different \( C \).
- Each member of the family of solutions satisfies the original differential equation.
- The concept emphasizes how varying initial conditions lead to different paths, visualizing a range of possibilities.
- A clear understanding of families of solutions helps in predicting behavior in dynamic systems.
Separation of Variables
Separation of variables is a method for solving certain types of differential equations by rearranging terms.The idea is to separate the equation into two parts, each containing only one variable.
In the example \( \frac{d y}{d t}=y^{2} \), by arranging terms, we arrive at \( \frac{1}{y^2} dy = dt \).This allows us to integrate both sides separately, ultimately finding \( -\frac{1}{y} = t + C \).
In the example \( \frac{d y}{d t}=y^{2} \), by arranging terms, we arrive at \( \frac{1}{y^2} dy = dt \).This allows us to integrate both sides separately, ultimately finding \( -\frac{1}{y} = t + C \).
- Separation of variables is particularly useful for first-order ordinary differential equations (ODEs) that can be written in a separable form.
- This technique simplifies equations, making them easier to solve analytically.
- Understanding this method is key in transforming complex real-world problems into manageable mathematical forms.
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