MU DIVISION

Question 21

Differential Equations★★★★★Question Bank

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Question

Let continuous, twice-differentiable functions

y_1(x)

y_2(x)

, and

y_3(x)

satisfy the following initial value problems:

y_1'+e^x\ln(x)y_1=0,\quad y_1(1)=2,

y_2'+\frac{e^x}{x}y_2=0,\quad y_2(1)=5,

y_3'+\frac{e^x}{x^2}y_3=0,\quad y_3(1)=10.

Evaluate:

\int_1^e\ln\left(\frac{y_1y_2^2}{y_3}\right)\,dx.

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Question

Let continuous, twice-differentiable functions

y_1(x)

y_2(x)

, and

y_3(x)

satisfy the following initial value problems:

y_1'+e^x\ln(x)y_1=0,\quad y_1(1)=2,

y_2'+\frac{e^x}{x}y_2=0,\quad y_2(1)=5,

y_3'+\frac{e^x}{x^2}y_3=0,\quad y_3(1)=10.

Evaluate:

\int_1^e\ln\left(\frac{y_1y_2^2}{y_3}\right)\,dx.