solve the initial value problems:
1- dq/dz = 2 + sinz, q(0)=5
2-dy/dx = (x^2 -2)x, y(2)=4
3-dy/dx = 4e^5^x + x, y(0)=4
4- dy/dx 3x + 2, y(3)= 2
To solve each of the given initial value problems, we will use an integrating factor and the initial condition to determine the constant of integration. Here's how to solve each problem step-by-step:
1. dq/dz = 2 + sin(z), q(0) = 5:
First, rewrite the differential equation as dq = (2 + sin(z))dz. Now, integrate both sides:
∫dq = ∫(2 + sin(z))dz
q = 2z - cos(z) + C
To find the value of the constant of integration, use the initial condition q(0) = 5:
5 = 2(0) - cos(0) + C
C = 7
Therefore, the solution to the initial value problem is q = 2z - cos(z) + 7.
2. dy/dx = (x^2 - 2)x, y(2) = 4:
Integrate both sides of the differential equation:
∫dy = ∫(x^2 - 2)x dx
y = (1/4)x^4 - (1/3)x^3 + C
Using the initial condition, y(2) = 4, substitute x = 2 and y = 4 to determine the constant of integration:
4 = (1/4)(2^4) - (1/3)(2^3) + C
C = 1
Hence, the solution to the initial value problem is y = (1/4)x^4 - (1/3)x^3 + 1.
3. dy/dx = 4e^(5x) + x, y(0) = 4:
Integrate both sides of the differential equation:
∫dy = ∫(4e^(5x) + x)dx
y = 4/5 * e^(5x) + (1/2)x^2 + C
Using the initial condition, y(0) = 4, substitute x = 0 and y = 4 to find the constant of integration:
4 = 4/5 * e^(5(0)) + (1/2)(0)^2 + C
C = 4 - 4/5
C = 16/5
Thus, the solution to the initial value problem is y = 4/5 * e^(5x) + (1/2)x^2 + 16/5.
4. dy/dx = 3x + 2, y(3) = 2:
Integrate both sides of the differential equation:
∫dy = ∫(3x + 2)dx
y = (3/2)x^2 + 2x + C
Using the initial condition, y(3) = 2, substitute x = 3 and y = 2 to determine the constant of integration:
2 = (3/2)(3^2) + 2(3) + C
C = 2 - (27/2) - 6
C = -19/2
Hence, the solution to the initial value problem is y = (3/2)x^2 + 2x - 19/2.