VIPSolutionsProvide answer without latex7.32 NE 82%u {) 25 app.outlierai/screening?id=68a5cc2 + H Outlier — 8 © /m 52s remaining Determine the general solution to the differential equation @ = 3+? - 4x. Provide a detailed explanation of the steps taken to integrate the function and the role of the constant of integration © ® 0 & I BE BN EE BE EE BE EE EE “111 vr 1 L111 “1 1 111111 2a 1 1 1 11 1 ES 1#1 , English(India) . J 9 Il @] bd
Question:
Provide answer without latex
7.32 NE 82%u
{) 25 app.outlierai/screening?id=68a5cc2 + H
Outlier — 8 © /m 52s remaining
Determine the general solution to the differential equation @ = 3+? - 4x. Provide a detailed explanation of the steps
taken to integrate the function and the role of the constant of integration
© ® 0 &
I BE BN EE BE EE BE EE EE
“111 vr 1 L111
“1 1 111111
2a 1 1 1 11 1 ES
1#1 , English(India) . J
9 Il @] bd
This is AI generated Response.
Asked by: Gurubindra Singh
Created at: 2025-08-20 19:33:43
Asked by: Gurubindra Singh
Created at: 2025-08-20 19:33:43
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Let's solve the differential equation:
dy/dx = 3x^2 - 4x
To find the general solution, we need to integrate both sides with respect to x.
Step 1: Write the equation in terms of differentials:
dy = (3x^2 - 4x) dx
Step 2: Integrate both sides:
∫ dy = ∫ (3x^2 - 4x) dx
Step 3: Integrate the right side term by term:
- The integral of 3x^2 is 3 times the integral of x^2, which is 3 * (x^3 / 3) = x^3
- The integral of -4x is -4 times the integral of x, which is -4 * (x^2 / 2) = -2x^2
So, after integrating:
y = x^3 - 2x^2 + C
Here, C is the constant of integration. This constant represents all the possible vertical shifts of the function, since any constant disappears when you take the derivative.
**Final Answer:**
The general solution is **y = x^3 - 2x^2 + C**.
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