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第4.問以下の微分方程式の一般解を求めよ。ただし，

y

は

x

の関数とする。 (1)

y^{\prime}=e^{x-y}+2

(2) $y^...

May 20, 2024

Solution by Steps

step 1

Rewrite the differential equation in the form

y'(x) = e^{x - y(x)} + 2

step 2

This is a first-order nonlinear ordinary differential equation

step 3

To solve, we use the integrating factor method or other suitable techniques for nonlinear ODEs

step 4

The general solution is

y(x) = c_1 e^{-x} + \frac{e^x}{2} + 2

Answer

y(x) = c_1 e^{-x} + \frac{e^x}{2} + 2

Key Concept

First-order nonlinear ODE

Explanation

This equation is solved using techniques for nonlinear differential equations, resulting in a general solution involving an exponential function.

Question 2:

y^{\prime}+y=e^{2 x}

step 1

Rewrite the differential equation in the form

y'(x) + y(x) = e^{2x}

step 2

This is a first-order linear ordinary differential equation

step 3

To solve, we use the integrating factor method

step 4

The general solution is

y(x) = c_1 e^{-x} + \frac{e^{2x}}{3}

Answer

y(x) = c_1 e^{-x} + \frac{e^{2x}}{3}

Key Concept

First-order linear ODE

Explanation

This equation is solved using the integrating factor method, resulting in a general solution involving an exponential function.

Question 3:

y^{\prime \prime}+5 y^{\prime}+4 y=0

step 1

Rewrite the differential equation in the form

y''(x) + 5y'(x) + 4y(x) = 0

step 2

This is a second-order linear homogeneous ordinary differential equation

step 3

To solve, we find the characteristic equation:

r^2 + 5r + 4 = 0

step 4

Solve the characteristic equation:

(r + 4)(r + 1) = 0

, giving roots

r = -4

and

r = -1

step 5

The general solution is

y(x) = c_1 e^{-4x} + c_2 e^{-x}

Answer

y(x) = c_1 e^{-4x} + c_2 e^{-x}

Key Concept

Second-order linear homogeneous ODE

Explanation

This equation is solved by finding the roots of the characteristic equation, resulting in a general solution involving exponential functions.