Question
upstudy study bank question image url

Solve the initial value problem: \( y^{\prime}(x)=\sqrt{4 y(x)+17}, \quad y(-2)=2 \) \( y(x)= \) Note: observe that the \( y(x) \) under the square root is not y times x . It is function notation, i.e. " y of x ".

Ask by Kelley Craig. in the United States
Mar 21,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The solution to the initial value problem is \( y(x) = \frac{(2x + 9)^2 - 17}{4} \).

Solution

We start with the differential equation \[ y'(x)=\sqrt{4\,y(x)+17}, \] and the initial condition \[ y(-2)=2. \] **Step 1. Separate variables** Write the differential equation as \[ \frac{dy}{dx}=\sqrt{4y+17}. \] Separate the variables by writing \[ \frac{dy}{\sqrt{4y+17}}=dx. \] **Step 2. Integrate both sides** Integrate the left-hand side with respect to \( y \) and the right-hand side with respect to \( x \): \[ \int \frac{dy}{\sqrt{4y+17}}=\int dx. \] **Step 3. Use a substitution for the left-hand side** Let \[ u=4y+17. \] Then \[ \frac{du}{dy}=4 \quad \Longrightarrow \quad dy=\frac{du}{4}. \] Substitute into the integral: \[ \int \frac{\frac{du}{4}}{\sqrt{u}}=\frac{1}{4}\int u^{-1/2}\,du. \] The integral is \[ \frac{1}{4}\int u^{-1/2}\,du = \frac{1}{4}\cdot 2u^{1/2} = \frac{1}{2}\sqrt{u} + C. \] Returning to the original variable \( y \), this becomes \[ \frac{1}{2}\sqrt{4y+17} + C. \] Thus, the integral equation is \[ \frac{1}{2}\sqrt{4y+17} = x + C_1, \] where \( C_1 \) is the constant of integration. **Step 4. Solve for \( y(x) \)** Multiply both sides by 2: \[ \sqrt{4y+17} = 2x + C_2, \] where \( C_2 = 2C_1 \). Square both sides to eliminate the square root: \[ 4y+17 = (2x+C_2)^2. \] Now, solve for \( y(x) \): \[ y(x) = \frac{(2x+C_2)^2-17}{4}. \] **Step 5. Apply the initial condition to determine \( C_2 \)** We know that \( y(-2)=2 \). Substitute \( x=-2 \) and \( y=2 \) into the equation before squaring (to avoid introducing an extraneous sign): From \[ \sqrt{4y+17}=2x+C_2, \] when \( x=-2 \) and \( y=2 \): \[ \sqrt{4\cdot2+17} = 2(-2)+C_2. \] Compute the left-hand side: \[ \sqrt{8+17}=\sqrt{25}=5. \] Then: \[ 5 = -4 + C_2 \quad \Longrightarrow \quad C_2=9. \] **Step 6. Write the final solution** Plug \( C_2=9 \) back into the equation for \( y(x) \): \[ \sqrt{4y+17}=2x+9. \] Square both sides: \[ 4y+17=(2x+9)^2. \] Solve for \( y(x) \): \[ y(x)=\frac{(2x+9)^2-17}{4}. \] Thus, the solution to the initial value problem is \[ \boxed{y(x)=\frac{(2x+9)^2-17}{4}}. \]

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

Mind Expander

To solve the initial value problem \( y^{\prime}(x) = \sqrt{4 y(x) + 17} \) with the initial condition \( y(-2) = 2 \), we start by separating the variables. 1. Rewrite the equation: \[ \frac{dy}{dx} = \sqrt{4y + 17} \] 2. Separate the variables \( y \) and \( x \): \[ \frac{dy}{\sqrt{4y + 17}} = dx \] 3. Now, let's integrate both sides. Starting with the left side: To integrate the left side, we can use the substitution method. Let \( u = 4y + 17 \), then \( du = 4 dy \) or \( dy = \frac{du}{4} \). Thus, \[ \int \frac{1}{\sqrt{u}} \cdot \frac{1}{4} du = \frac{1}{4} \cdot 2\sqrt{u} + C = \frac{1}{2} \sqrt{4y + 17} + C \] And the right side is simply: \[ \int dx = x + C_1 \] 4. Putting this together, we have: \[ \frac{1}{2} \sqrt{4y + 17} = x + C \] 5. Multiplying both sides by 2: \[ \sqrt{4y + 17} = 2x + 2C \] 6. Squaring both sides gives: \[ 4y + 17 = (2x + 2C)^2 \] 7. Simplifying leads to: \[ 4y = 4x^2 + 8Cx + 4C^2 - 17 \] \[ y = x^2 + 2Cx + C^2 - \frac{17}{4} \] 8. Now, we will use the initial condition \( y(-2) = 2 \) to find \( C \): \[ 2 = (-2)^2 + 2C(-2) + C^2 - \frac{17}{4} \] \[ 2 = 4 - 4C + C^2 - \frac{17}{4} \] This simplifies to: \[ 2 = \frac{16}{4} - 4C + C^2 - \frac{17}{4} \] \[ 2 = -4C + C^2 - \frac{1}{4} \] Multiply through by 4 to eliminate the fraction: \[ 8 = -16C + 4C^2 - 1 \] \[ 4C^2 - 16C - 9 = 0 \] Using the quadratic formula, \( C = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): \[ C = \frac{16 \pm \sqrt{256 + 144}}{8} = \frac{16 \pm 20}{8} \] So, \( C = 4.5 \) or \( C = -0.5 \). 9. We would take the appropriate \( C \) from the context of the problem (usually the positive constant would be preferred unless a particular behavior is needed). 10. Assuming \( C = -0.5 \), we have: \[ y = x^2 - (2)(0.5)x + 0.25 - \frac{17}{4} \] Thus upon simplifying yields: \[ y = x^2 - x - \frac{15}{4} \] Therefore, the solution to the initial value problem is: \[ y(x) = x^2 - x - \frac{15}{4} \]

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy