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Magazine Problem 3
Obtain a power series solution in powers of \(x\) of each of the initial-value problems by (a) the Taylor series method and (b) the method of undetermined coefficients. $$ \frac{d y}{d x}=1+x y^{2}, \quad y(0)=2 $$
Problem 4
For each of the initial-value problems use the method of successive approximations to find the first three members \(\phi_{1}, \phi_{2}, \phi_{3}\) of a sequence of functions that approaches the exact solution of the problem. $$ \frac{d y}{d x}=1+x y^{2}, \quad y(0)=0 $$
Problem 4
Employ the method of isoclines to sketch the approximate integral curves of each of the differential equations $$ \frac{d y}{d x}=x^{2}+2 y^{2} $$
Problem 4
Obtain a power series solution in powers of \(x\) of each of the initial-value problems by (a) the Taylor series method and (b) the method of undetermined coefficients. $$ \frac{d y}{d x}=x^{3}+y^{3}, \quad y(0)=3 $$
Problem 5
Employ the method of isoclines to sketch the approximate integral curves of each of the differential equations $$ \frac{d y}{d x}=\frac{3 x-y}{x+y} $$
Problem 5
For each of the initial-value problems use the method of successive approximations to find the first three members \(\phi_{1}, \phi_{2}, \phi_{3}\) of a sequence of functions that approaches the exact solution of the problem. $$ \frac{d y}{d x}=e^{x}+y^{2}, \quad y(0)=0 $$
Problem 5
Obtain a power series solution in powers of \(x\) of each of the initial-value problems by (a) the Taylor series method and (b) the method of undetermined coefficients. $$ \frac{d y}{d x}=x+\sin y, \quad y(0)=0 $$
Problem 6
Employ the method of isoclines to sketch the approximate integral curves of each of the differential equations $$ \frac{d y}{d x}=\sin x-y $$
Problem 6
For each of the initial-value problems use the method of successive approximations to find the first three members \(\phi_{1}, \phi_{2}, \phi_{3}\) of a sequence of functions that approaches the exact solution of the problem. $$ \frac{d y}{d x}=\sin x+y^{2}, \quad y(0)=0 $$
Problem 6
Obtain a power series solution in powers of \(x\) of each of the initial-value problems by (a) the Taylor series method and (b) the method of undetermined coefficients. $$ \frac{d y}{d x}=1+x \sin y, \quad y(0)=0 $$
