Question Statement

Solve the differential equation:

$2 (y - 1) d y = (3 x^{2} + 4 x + 2) d x$

with the initial condition $y (0) = - 1$ .

Background and Explanation

This problem involves a separable differential equation, where we can rearrange the equation so that all terms involving $y$ are on one side and all terms involving $x$ are on the other. Once separated, we integrate both sides independently and use the initial condition to determine the constant of integration.

Solution

First, expand the left-hand side to prepare for integration:

$2 (y - 1) d y = (2 y - 2) d y$

Now our equation is:

$(2 y - 2) d y = (3 x^{2} + 4 x + 2) d x$

Integrate both sides:

$\int (2 y - 2) d y = \int (3 x^{2} + 4 x + 2) d x$

Apply the linearity property of integrals to break this into simpler integrals:

$2 \int y d y - 2 \int 1 d y = 3 \int x^{2} d x + 4 \int x d x + 2 \int 1 d x$

Evaluate each integral:

$2 \cdot \frac{y ^{2}}{2} - 2 y = 3 \cdot \frac{x ^{3}}{3} + 4 \cdot \frac{x ^{2}}{2} + 2 x + c$

Simplify by cancelling coefficients (the 2s on the left, and the 3 and 4 on the right with their respective denominators):

$y^{2} - 2 y = x^{3} + 2 x^{2} + 2 x + c$

Apply the initial condition $y (0) = - 1$ :

When $x = 0$ , we have $y = - 1$ . Substitute these values into equation (1):

$(- 1)^{2} - 2 (- 1) = (0)^{3} + 2 (0)^{2} + 2 (0) + c$

Simplify:

$1 + 2 = 0 + 0 + 0 + c$

$3 = c$

Substitute $c = 3$ back into equation (1):

$y^{2} - 2 y = x^{3} + 2 x^{2} + 2 x + 3$

This is the particular solution to the differential equation satisfying the given initial condition.

Key Formulas or Methods Used

Separation of variables: Rearranging a differential equation so that each variable appears on only one side
Power rule for integration: $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C$ for $n \neq = - 1$
Linearity of integration: $\int [a f (x) + b g (x)] d x = a \int f (x) d x + b \int g (x) d x$
Initial value problems: Using a given point $(x_{0}, y_{0})$ to solve for the constant of integration

Summary of Steps

Expand and separate: Rewrite $2 (y - 1) d y$ as $(2 y - 2) d y$ to match the separated form
Integrate both sides: Compute $\int (2 y - 2) d y = \int (3 x^{2} + 4 x + 2) d x$ to obtain $y^{2} - 2 y = x^{3} + 2 x^{2} + 2 x + c$
Apply initial condition: Substitute $x = 0$ and $y = - 1$ to get $1 + 2 = c$ , so $c = 3$
State final answer: Substitute $c = 3$ to get the particular solution $y^{2} - 2 y = x^{3} + 2 x^{2} + 2 x + 3$

Question Statement

Solve the differential equation:

$2 (y - 1) d y = (3 x^{2} + 4 x + 2) d x$

with the initial condition $y (0) = - 1$ .

Background and Explanation

Solution

First, expand the left-hand side to prepare for integration:

$2 (y - 1) d y = (2 y - 2) d y$

Now our equation is:

$(2 y - 2) d y = (3 x^{2} + 4 x + 2) d x$

Integrate both sides:

$\int (2 y - 2) d y = \int (3 x^{2} + 4 x + 2) d x$

Apply the linearity property of integrals to break this into simpler integrals:

$2 \int y d y - 2 \int 1 d y = 3 \int x^{2} d x + 4 \int x d x + 2 \int 1 d x$

Evaluate each integral:

$2 \cdot \frac{y ^{2}}{2} - 2 y = 3 \cdot \frac{x ^{3}}{3} + 4 \cdot \frac{x ^{2}}{2} + 2 x + c$

Simplify by cancelling coefficients (the 2s on the left, and the 3 and 4 on the right with their respective denominators):

$y^{2} - 2 y = x^{3} + 2 x^{2} + 2 x + c$

Apply the initial condition $y (0) = - 1$ :

When $x = 0$ , we have $y = - 1$ . Substitute these values into equation (1):

$(- 1)^{2} - 2 (- 1) = (0)^{3} + 2 (0)^{2} + 2 (0) + c$

Simplify:

$1 + 2 = 0 + 0 + 0 + c$

$3 = c$

Substitute $c = 3$ back into equation (1):

$y^{2} - 2 y = x^{3} + 2 x^{2} + 2 x + 3$

This is the particular solution to the differential equation satisfying the given initial condition.

Key Formulas or Methods Used

Separation of variables: Rearranging a differential equation so that each variable appears on only one side
Power rule for integration: $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C$ for $n \neq = - 1$
Linearity of integration: $\int [a f (x) + b g (x)] d x = a \int f (x) d x + b \int g (x) d x$
Initial value problems: Using a given point $(x_{0}, y_{0})$ to solve for the constant of integration

Summary of Steps

Expand and separate: Rewrite $2 (y - 1) d y$ as $(2 y - 2) d y$ to match the separated form
Integrate both sides: Compute $\int (2 y - 2) d y = \int (3 x^{2} + 4 x + 2) d x$ to obtain $y^{2} - 2 y = x^{3} + 2 x^{2} + 2 x + c$
Apply initial condition: Substitute $x = 0$ and $y = - 1$ to get $1 + 2 = c$ , so $c = 3$
State final answer: Substitute $c = 3$ to get the particular solution $y^{2} - 2 y = x^{3} + 2 x^{2} + 2 x + 3$

4.2 Q-15

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

4.2 Q-15

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps