Question

Find the particular solution to the differential equation $$ y^{\prime}=4 x+3 $$ passing through the point \((1,7)\), given that \(y=2 x^{2}+3 x+C\) is a general solution to the differential equation.

Short answer

The particular solution is \( y = 2x^2 + 3x + 2 \).

Step-by-step solution

Understand the Given Information

The problem provides a differential equation \( y' = 4x + 3 \) and a general solution \( y = 2x^2 + 3x + C \). We are tasked with finding a particular solution passing through the point \((1, 7)\). The particular solution will involve finding the specific value of \(C\) that satisfies this condition.

Verify General Solution Satisfies Differential Equation

Differentiate the given general solution \( y = 2x^2 + 3x + C \) to find \( y' \). Compute \( y' = 4x + 3 \), which matches the given differential equation. This confirms \( y = 2x^2 + 3x + C \) is indeed a general solution.

Substitute the Point into the General Solution

The point \((1, 7)\) is on the curve described by the solution. Substitute \(x = 1\) and \(y = 7\) into the equation \( y = 2x^2 + 3x + C \). This yields the equation \( 7 = 2(1)^2 + 3(1) + C \).

Solve for the Constant C

Simplify the equation from Step 3: \( 7 = 2 + 3 + C \), which simplifies to \( 7 = 5 + C \). Solve for \( C \) by subtracting 5 from both sides, resulting in \( C = 2 \).

Write the Particular Solution

Substitute the value of \( C \) into the general solution to find the particular solution: \( y = 2x^2 + 3x + 2 \). This function passes through the point \((1, 7)\).

Key concepts

General Solution

The general solution of a differential equation is a solution that encompasses all possible solutions of the equation. It includes unknown constants, which makes it adjustable to fit specific scenarios.
In our problem, the general solution is given as \( y = 2x^2 + 3x + C \). The inclusion of \( C \) in the solution means that it's a family of curves — each specific curve is determined by a specific value of \( C \).
This general solution fits the differential equation \( y' = 4x + 3 \), as differentiating \( y = 2x^2 + 3x + C \) results in \( y' = 4x + 3 \), which matches the differential equation given in the problem. Thus, confirming that the expression \( y = 2x^2 + 3x + C \) is indeed a valid general solution.

Particular Solution

A particular solution is a specific form of the general solution that satisfies a given condition, like passing through a specific point.
In our case, the task is to find a particular solution that passes through the point \((1, 7)\). This involves substituting the point into the general solution \( y = 2x^2 + 3x + C \) as follows:

• Substitute \( x = 1 \) and \( y = 7 \) to get the equation \( 7 = 2(1)^2 + 3(1) + C \).
• Simplify this to find \( C = 2 \).

This leads to the particular solution \( y = 2x^2 + 3x + 2 \), which satisfies both the differential equation and the point \((1, 7)\).

Initial Condition

Initial conditions are specific values provided to determine the constants in the general solution and find the particular solution.
They are crucial as they narrow down the infinite possibilities of a general solution to a single solution that passes through a given point or fulfills a specific requirement.
In this example, the initial condition was given in the form of the point \((1, 7)\).
By substituting this into the general solution, we calculated the constant \( C = 2 \).
This condition ensures that the particular solution \( y = 2x^2 + 3x + 2 \) is uniquely determined and accurate for the situation described.

Constant of Integration

The constant of integration, denoted here as \( C \), is an arbitrary constant that arises when integrating differential equations.
When finding the general solution to a differential equation, this constant represents an infinite number of potential solutions. Each different value of \( C \) corresponds to a different solution, or curve, that satisfies the differential equation.

• In our problem, \( C \) begins as an unknown in the general solution: \( y = 2x^2 + 3x + C \).
• Only when the initial condition \((1, 7)\) is applied does \( C \) take on a specific value of \( C = 2 \).

The constant of integration showcases how initial conditions are crucial for narrowing down the infinite possibilities offered by the general solution.